:: TAXONOM2 semantic presentation
:: deftheorem Def1 defines with_superior TAXONOM2:def 1 :
:: deftheorem Def2 defines with_comparable_down TAXONOM2:def 2 :
theorem Th1: :: TAXONOM2:1
theorem Th2: :: TAXONOM2:2
theorem Th3: :: TAXONOM2:3
theorem Th4: :: TAXONOM2:4
theorem Th5: :: TAXONOM2:5
:: deftheorem Def3 defines hierarchic TAXONOM2:def 3 :
for b
1 being
set holds
( b
1 is
hierarchic iff for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & not b
2 c= b
3 & not b
3 c= b
2 & not b
2 misses b
3 ) );
theorem Th6: :: TAXONOM2:6
theorem Th7: :: TAXONOM2:7
:: deftheorem Def4 defines Hierarchy TAXONOM2:def 4 :
:: deftheorem Def5 defines mutually-disjoint TAXONOM2:def 5 :
theorem Th8: :: TAXONOM2:8
theorem Th9: :: TAXONOM2:9
theorem Th10: :: TAXONOM2:10
E12:
now
let c
1 be
set ;
let c
2 be
Hierarchy of c
1;
assume E13:
c
2 is
covering
;
let c
3 be
mutually-disjoint Subset-Family of c
1;
assume E14:
( c
3 c= c
2 & ( for b
1 being
mutually-disjoint Subset-Family of c
1 holds
( b
1 c= c
2 &
union c
3 c= union b
1 implies c
3 = b
1 ) ) )
;
thus
union c
3 = c
1
proof
thus
union c
3 c= c
1
;
:: according to XBOOLE_0:def 10
thus
c
1 c= union c
3
proof
let c
4 be
set ;
:: according to TARSKI:def 3
assume E15:
c
4 in c
1
;
c
4 in union c
2
by E13, E15, ABIAN:4;
then consider c
5 being
set such that E16:
( c
4 in c
5 & c
5 in c
2 )
by TARSKI:def 4;
now
assume E17:
not c
4 in union c
3
;
defpred S
1[
set ] means a
1 c= c
5;
consider c
6 being
set such that E18:
for b
1 being
set holds
( b
1 in c
6 iff ( b
1 in c
3 & S
1[b
1] ) )
from XBOOLE_0:sch 1();
set c
7 =
(c3 \ c6) \/ {c5};
c
5 in {c5}
by TARSKI:def 1;
then E19:
c
5 in (c3 \ c6) \/ {c5}
by XBOOLE_0:def 2;
E20:
c
3 \ c
6 c= (c3 \ c6) \/ {c5}
by XBOOLE_1:7;
c
3 \ c
6 c= c
3
by XBOOLE_1:36;
then E21:
c
3 \ c
6 c= c
2
by E14, XBOOLE_1:1;
{c5} c= c
2
then E22:
(c3 \ c6) \/ {c5} c= c
2
by E21, XBOOLE_1:8;
then E23:
(c3 \ c6) \/ {c5} c= bool c
1
by XBOOLE_1:1;
E24:
for b
1 being
set holds
( b
1 in c
3 & not b
1 in c
6 & b
1 <> c
5 implies b
1 misses c
5 )
for b
1, b
2 being
set holds
( b
1 in (c3 \ c6) \/ {c5} & b
2 in (c3 \ c6) \/ {c5} & b
1 <> b
2 implies b
1 misses b
2 )
then E25:
(c3 \ c6) \/ {c5} is
mutually-disjoint Subset-Family of c
1
by E23, Def5;
union c
3 c= union ((c3 \ c6) \/ {c5})
then E26:
c
3 = (c3 \ c6) \/ {c5}
by E14, E22, E25;
E27:
{c5} c= (c3 \ c6) \/ {c5}
by XBOOLE_1:7;
c
5 in {c5}
by TARSKI:def 1;
hence
not verum
by E16, E17, E26, E27, TARSKI:def 4;
end;
hence
c
4 in union c
3
;
end;
end;
end;
:: deftheorem Def6 defines T_3 TAXONOM2:def 6 :
for b
1 being
set for b
2 being
Subset-Family of b
1 holds
( b
2 is
T_3 iff for b
3 being
Subset of b
1 holds
( b
3 in b
2 implies for b
4 being
Element of b
1 holds
not ( not b
4 in b
3 & ( for b
5 being
Subset of b
1 holds
not ( b
4 in b
5 & b
5 in b
2 & b
3 misses b
5 ) ) ) ) );
theorem Th11: :: TAXONOM2:11
:: deftheorem Def7 defines lower-bounded TAXONOM2:def 7 :
theorem Th12: :: TAXONOM2:12
:: deftheorem Def8 defines with_max's TAXONOM2:def 8 :
theorem Th13: :: TAXONOM2:13
theorem Th14: :: TAXONOM2:14
theorem Th15: :: TAXONOM2:15
theorem Th16: :: TAXONOM2:16
theorem Th17: :: TAXONOM2:17
theorem Th18: :: TAXONOM2:18
theorem Th19: :: TAXONOM2:19
theorem Th20: :: TAXONOM2:20