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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">c</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>   <a href="aofa_000.html#V1" title="AOFA_000:attr.1">associative</a>   <a href="aofa_000.html#V2" title="AOFA_000:attr.2">unital</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Olive" title="b2">c</font> holds <br/>for <font color="Olive" title="b3">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">w</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b5">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b2">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b5">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b5">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>   <a href="aofa_000.html#V1" title="AOFA_000:attr.1">associative</a>   <a href="aofa_000.html#V2" title="AOFA_000:attr.2">unital</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">x</font> <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Olive" title="b1">c</font> holds <br/>for <font color="Olive" title="b2">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">w</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b4">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Olive" title="b1">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b4">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">c</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>   <a href="aofa_000.html#V1" title="AOFA_000:attr.1">associative</a>   <a href="aofa_000.html#V2" title="AOFA_000:attr.2">unital</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</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="c2">x</font> <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Maroon" title="c3">c</font> implies for <font color="Olive" title="b1">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">w</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b3">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:69"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">w</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b3">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">w</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b2">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">w</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">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">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c6">Y</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">e</font> = <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">e</font> = <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V8" title="COMPUT_1:attr.8">nullary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">char</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">ec</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E4:69"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> is   <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">PFuncFinSequence</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</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="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</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="c11">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> or  not <font color="Maroon" title="c11">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/>
<font color="Maroon" title="c11">a</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"><span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
 ; <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="c11">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E3:69_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">a</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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> or <font color="Maroon" title="c11">a</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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:69_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></span>;<br/>
<a NAME="E5:69_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:69_1"/>
not <font color="Maroon" title="c11">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">char</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">PFuncFinSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">S</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c11">char</font> #) as   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D9" title="UNIALG_1:def.9">UNIALG_1:def 9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E7:69"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:69"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></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="c12">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<a NAME="E9:69"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E10:69"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E11:69"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">char</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<a NAME="E12:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">char</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">e</font> = <font color="Maroon" title="c8">e</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="comput_1.html#V8" title="COMPUT_1:attr.8">nullary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c3">c</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>   <a href="aofa_000.html#V1" title="AOFA_000:attr.1">associative</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> ;<br/>
<a NAME="E15:69"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E16:69"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E17:69"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E18:69"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<span class="kw">thus </span><a NAME="E19:69"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_000.html#D10" target="_self" title="AOFA_000:def.10">AOFA_000:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V8" title="COMPUT_1:attr.8">nullary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<a NAME="E20:69"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E21:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A9</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
;<br/>
<a NAME="E22:69"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:69"><span class="lab"><font color="Green" title="E53">A8</font></span></a></span>;<br/>
<a NAME="E23:69"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:69"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E21:69"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E24:69"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E22:69"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E25:69"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V8" title="COMPUT_1:attr.8">nullary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:69"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E21:69"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></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="c12">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E26:69"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_000.html#D11" target="_self" title="AOFA_000:def.11">AOFA_000:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E27:69"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E22:69"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></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="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E28:69"/>
<font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</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="c12">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c14">c</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_000.html#D15" target="_self" title="AOFA_000:def.15">AOFA_000:def 15</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="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Maroon" title="c14">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:69_2"/>
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E22:69"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Maroon" title="c14">c</font></span><br/>

<a NAME="E2:69_2"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>;<br/>
<a NAME="E3:69_2"/><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T8" title="COMPUT_1:th.8">COMPUT_1:8</a></span>;<br/>
<a NAME="E4:69_2"/><span class="lab"><font color="Green" title="E60">A15</font></span>: 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_2"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/>
<a NAME="E5:69_2"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:69_2"/>
<font color="Maroon" title="c13">e</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:69"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E21:69"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E4:69_2"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:69_2"/>
<span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Maroon" title="c14">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:69_2"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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">thus </span><a NAME="E29:69"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>   <a href="aofa_000.html#V1" title="AOFA_000:attr.1">associative</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:69"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E22:69"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_000.html#D14" target="_self" title="AOFA_000:def.14">AOFA_000:def 14</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="c12">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E30:69"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_000.html#D12" target="_self" title="AOFA_000:def.12">AOFA_000:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> &amp; <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<a NAME="E31:69"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E32:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A18</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
;<br/>
<a NAME="E33:69"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:69"><span class="lab"><font color="Green" title="E58">A17</font></span></a></span>;<br/>
<a NAME="E34:69"/><span class="lab"><font color="Green" title="E61">A20</font></span>: 
<font color="Maroon" title="c11">char</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E17:69"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E32:69"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E35:69"/><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c11">char</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E18:69"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E33:69"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E36:69"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:69"><span class="lab"><font color="Green" title="E61">A20</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"> ( <font color="Maroon" title="c12">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E37:69"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aofa_000.html#D13" target="_self" title="AOFA_000:def.13">AOFA_000:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E38:69"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E18:69"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E33:69"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></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="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E39:69"/>
<font color="Maroon" title="c12">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</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="c12">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="unialg_1.html#D5" title="UNIALG_1:def.5">UNIALG_1:def 5</a>,<a class="ref" href="unialg_1.html#D8" title="UNIALG_1:def.8">UNIALG_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  holds <br/>( <font color="Maroon" title="c15">i</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> or <font color="Olive">b<sub>1</sub></font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</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="c15">i</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> or  not <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> or <font color="Maroon" title="c16">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:69_3"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 ; <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="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> or <font color="Maroon" title="c16">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:69_3"/>
(  not <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> or <font color="Maroon" title="c16">h</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E23:69"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E24:69"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E34:69"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="txt" href="#E35:69"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a>, <a class="ref" href="finseq_3.html#T2" title="FINSEQ_3:th.2">FINSEQ_3:2</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">let </span><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a>,<a class="ref" href="unialg_1.html#D7" title="UNIALG_1:def.7">UNIALG_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  holds <br/>( <font color="Maroon" title="c15">i</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> or <font color="Olive">b<sub>1</sub></font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">S</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="c15">i</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> or  not <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> or <font color="Maroon" title="c16">h</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E40:69"/><span class="lab"><font color="Green" title="E63">A22</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font>
 <span class="kw">and </span><br/><a NAME="E41:69"/><span class="lab"><font color="Green" title="E64">A23</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c12">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</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="c16">h</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c17">p1</font>, <font color="Maroon" title="c18">p2</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="unialg_1.html#D1" title="UNIALG_1:def.1">UNIALG_1:def 1</a>,<a class="ref" href="margrel1.html#D1" title="MARGREL1:def.1">MARGREL1:def 1</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="c17">p1</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c16">h</font> or <font color="Maroon" title="c18">p2</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c16">h</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</font> )</span><br/>

<a NAME="E42:69"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E40:69"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a>, <a class="ref" href="finseq_3.html#T2" title="FINSEQ_3:th.2">FINSEQ_3:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E43:69"/>
( <font color="Maroon" title="c17">p1</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c16">h</font> or <font color="Maroon" title="c18">p2</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c16">h</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:69"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E24:69"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E34:69"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="txt" href="#E35:69"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="txt" href="#E41:69"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="margrel1.html#D1" title="MARGREL1:def.1">MARGREL1:def 1</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>
