:: Propositional Calculus for Boolean Valued Functions, II
:: by Shunichi Kobayashi and Yatsuka Nakamura
::
:: Received March 13, 1999
:: Copyright (c) 1999-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XBOOLE_0, SUBSET_1, MARGREL1, BVFUNC_1, XBOOLEAN, FUNCT_1,
PARTIT1;
notations XBOOLE_0, SUBSET_1, XBOOLEAN, MARGREL1, FUNCT_2, BVFUNC_1;
constructors BINARITH, BVFUNC_1, XBOOLEAN, NUMBERS;
registrations XBOOLEAN, MARGREL1, ORDINAL1;
requirements ARITHM, NUMERALS;
definitions BVFUNC_1, FUNCT_2;
equalities XBOOLEAN;
expansions BVFUNC_1;
theorems MARGREL1, BINARITH, BVFUNC_1, XBOOLEAN, BVFUNC_4, BVFUNC_5;
begin
reserve Y for non empty set;
theorem
for a,b being Function of Y,BOOLEAN holds a 'imp' (b 'imp' (a
'&' b))=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds (a 'imp' (b 'imp' (a '&' b))).x=TRUE
proof
let x be Element of Y;
(a 'imp' (b 'imp' (a '&' b))).x ='not' a.x 'or' (b 'imp' (a '&' b)).
x by BVFUNC_1:def 8
.='not' a.x 'or' ('not' b.x 'or' (a '&' b).x) by BVFUNC_1:def 8
.='not' a.x 'or' ('not' b.x 'or' (a.x '&' b.x)) by
MARGREL1:def 20
.='not' a.x 'or' (('not' b.x 'or' a.x) '&' ('not' b.x 'or' (b)
.x)) by XBOOLEAN:9
.='not' a.x 'or' (TRUE '&' ('not' b.x 'or' a.x)) by XBOOLEAN:102
.=('not' a.x 'or' a.x) 'or' 'not' b.x
.=TRUE 'or' 'not' b.x by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((b
'imp' a) 'imp' (a 'eqv' b))=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a
'eqv' b))).x=TRUE
proof
let x be Element of Y;
((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))).x ='not' (a 'imp'
b).x 'or' ((b 'imp' a) 'imp' (a 'eqv' b)).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' b.x) 'or' ((b 'imp' a) 'imp' (a 'eqv' b)).
x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' b.x) 'or' ('not' (b 'imp' a).x 'or' (a
'eqv' b).x) by BVFUNC_1:def 8
.=('not' 'not' a.x '&' 'not' b.x) 'or' ('not'( 'not' b.x 'or' (a
).x) 'or' (a 'eqv' b).x) by BVFUNC_1:def 8
.=(a.x '&' 'not' b.x) 'or' ((b.x '&' 'not' a.x) 'or' 'not'( (a
).x 'xor' b.x)) by BVFUNC_1:def 9
.=(a.x '&' 'not' b.x) 'or' ((('not' a.x '&' b.x) 'or' 'not'(
'not' a.x '&' b.x)) '&' (('not' a.x '&' b.x) 'or' 'not'( a.x '&'
'not' b.x))) by XBOOLEAN:9
.=(a.x '&' 'not' b.x) 'or' (TRUE '&' (('not' a.x '&' b.x) 'or'
'not'( a.x '&' 'not' b.x))) by XBOOLEAN:102
.=((a.x '&' 'not' b.x) 'or' 'not'( a.x '&' 'not' b.x)) 'or' (
'not' a.x '&' b.x)
.=TRUE 'or' ('not' a.x '&' b.x) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'or' b) 'eqv' (b
'or' a)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'or' b) 'eqv' (b 'or' a)).x=TRUE
proof
let x be Element of Y;
((a 'or' b) 'eqv' (b 'or' a)).x ='not'( (a 'or' b).x 'xor' (b 'or' a).
x) by BVFUNC_1:def 9
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' c)
'imp' (a 'imp' (b 'imp' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a '&' b) 'imp' c) 'imp' (a 'imp' (b
'imp' c))).x=TRUE
proof
let x be Element of Y;
(((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))).x ='not' ((a '&' b)
'imp' c).x 'or' (a 'imp' (b 'imp' c)).x by BVFUNC_1:def 8
.='not'( 'not' (a '&' b).x 'or' c.x) 'or' (a 'imp' (b 'imp' c)).x by
BVFUNC_1:def 8
.='not'( 'not' (a.x '&' b.x) 'or' c.x) 'or' (a 'imp' (b 'imp' c)
).x by MARGREL1:def 20
.='not'( 'not'( a.x '&' b.x) 'or' c.x) 'or' ('not' a.x 'or' (b
'imp' c).x) by BVFUNC_1:def 8
.='not'( ('not' a.x 'or' 'not' b.x) 'or' c.x) 'or' ('not' a.x
'or' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c)
) 'imp' ((a '&' b) 'imp' c)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'imp' (b 'imp' c)) 'imp' ((a '&' b)
'imp' c)).x=TRUE
proof
let x be Element of Y;
((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)).x ='not' (a 'imp' (b
'imp' c)).x 'or' ((a '&' b) 'imp' c).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' (b 'imp' c).x) 'or' ((a '&' b) 'imp' c).x by
BVFUNC_1:def 8
.='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ((a '&' b)
'imp' c).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ('not' (a '&'
b).x 'or' c.x) by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' (('not' a.x
'or' 'not' b.x) 'or' c.x) by MARGREL1:def 20
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a) 'imp' ((
c 'imp' b) 'imp' (c 'imp' (a '&' b)))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds ((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c
'imp' (a '&' b)))).x=TRUE
proof
let x be Element of Y;
((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))).x ='not' (
c 'imp' a).x 'or' ((c 'imp' b) 'imp' (c 'imp' (a '&' b))).x by BVFUNC_1:def 8
.='not'( 'not' c.x 'or' a.x) 'or' ((c 'imp' b) 'imp' (c 'imp' (a
'&' b))).x by BVFUNC_1:def 8
.='not'( 'not' c.x 'or' a.x) 'or' ('not' (c 'imp' b).x 'or' (c
'imp' (a '&' b)).x) by BVFUNC_1:def 8
.='not'( 'not' c.x 'or' a.x) 'or' ('not'( 'not' c.x 'or' b.x)
'or' (c 'imp' (a '&' b)).x) by BVFUNC_1:def 8
.='not'( 'not' c.x 'or' a.x) 'or' ('not'( 'not' c.x 'or' b.x)
'or' ('not' c.x 'or' (a '&' b).x)) by BVFUNC_1:def 8
.=(c.x '&' 'not' a.x) 'or' (('not' 'not' c.x '&' 'not' b.x)
'or' ('not' c.x 'or' (a.x '&' b.x))) by MARGREL1:def 20
.=(c.x '&' 'not' a.x) 'or' ((c.x '&' 'not' b.x) 'or' (('not' (
c).x 'or' a.x) '&' ('not' c.x 'or' b.x))) by XBOOLEAN:9
.=(c.x '&' 'not' a.x) 'or' (((c.x '&' 'not' b.x) 'or' ('not' (
c).x 'or' a.x)) '&' ((c.x '&' 'not' b.x) 'or' ('not' c.x 'or' 'not'
'not' b.x))) by XBOOLEAN:9
.=(c.x '&' 'not' a.x) 'or' (TRUE '&' ((c.x '&' 'not' b.x) 'or'
('not' c.x 'or' a.x))) by XBOOLEAN:102
.=((c.x '&' 'not' a.x) 'or' 'not'( c.x '&' 'not' a.x)) 'or' ((
c).x '&' 'not' b.x)
.=TRUE 'or' (c.x '&' 'not' b.x) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) 'imp' c)
'imp' ((a 'imp' c) 'or' (b 'imp' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'or' b) 'imp' c) 'imp' ((a 'imp' c)
'or' (b 'imp' c))).x=TRUE
proof
let x be Element of Y;
(((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))).x ='not' (
(a 'or' b) 'imp' c).x 'or' ((a 'imp' c) 'or' (b 'imp' c)).x by BVFUNC_1:def 8
.='not'( 'not' (a 'or' b).x 'or' c.x) 'or' ((a 'imp' c) 'or' (b
'imp' c)).x by BVFUNC_1:def 8
.='not'( 'not'( a.x 'or' b.x) 'or' c.x) 'or' ((a 'imp' c) 'or' (
b 'imp' c)).x by BVFUNC_1:def 4
.='not'( 'not'( a.x 'or' b.x) 'or' c.x) 'or' ((a 'imp' c).x 'or'
(b 'imp' c).x) by BVFUNC_1:def 4
.='not'( 'not'( a.x 'or' b.x) 'or' c.x) 'or' (('not' a.x 'or'
c.x) 'or' (b 'imp' c).x) by BVFUNC_1:def 8
.=('not' 'not'( a.x 'or' b.x) '&' 'not' c.x) 'or' (('not' a.x
'or' c.x) 'or' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8
.=((b.x '&' 'not' c.x) 'or' (a.x '&' 'not' c.x)) 'or' ('not'(
a.x '&' 'not' c.x) 'or' ('not' b.x 'or' c.x)) by XBOOLEAN:8
.=((b.x '&' 'not' c.x) 'or' ((a.x '&' 'not' c.x) 'or' 'not'( (
a).x '&' 'not' c.x))) 'or' ('not' b.x 'or' c.x)
.=((b.x '&' 'not' c.x) 'or' TRUE) 'or' ('not' b.x 'or' c.x) by
XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c) 'imp' ((
b 'imp' c) 'imp' ((a 'or' b) 'imp' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a
'or' b) 'imp' c))).x=TRUE
proof
let x be Element of Y;
((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))).x ='not'
(a 'imp' c).x 'or' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' c.x) 'or' ((b 'imp' c) 'imp' ((a 'or' b)
'imp' c)).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' c.x) 'or' ('not' (b 'imp' c).x 'or' ((a
'or' b) 'imp' c).x) by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x)
'or' ((a 'or' b) 'imp' c).x) by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x)
'or' ('not' (a 'or' b).x 'or' c.x)) by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x)
'or' ('not' (a.x 'or' b.x) 'or' c.x)) by BVFUNC_1:def 4
.='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x)
'or' ((c.x 'or' 'not' a.x) '&' ('not' b.x 'or' c.x))) by XBOOLEAN:9
.='not'( 'not' a.x 'or' c.x) 'or' (('not'( 'not' b.x 'or' c.x)
'or' (c.x 'or' 'not' a.x)) '&' ('not'( 'not' b.x 'or' c.x) 'or' ('not'
b.x 'or' c.x))) by XBOOLEAN:9
.='not'( 'not' a.x 'or' c.x) 'or' (TRUE '&' ('not'( 'not' b.x
'or' c.x) 'or' (c.x 'or' 'not' a.x))) by XBOOLEAN:102
.='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x)
'or' ('not' a.x 'or' c.x))
.=('not'( 'not' a.x 'or' c.x) 'or' ('not' a.x 'or' c.x)) 'or'
'not'( 'not' b.x 'or' c.x)
.=TRUE 'or' 'not'( 'not' b.x 'or' c.x) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem Th9:
for a,b,c being Function of Y,BOOLEAN holds ((a 'imp' c) '&' (b
'imp' c)) 'imp' ((a 'or' b) 'imp' c)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a
'or' b) 'imp' c)).x=TRUE
proof
let x be Element of Y;
(((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)).x ='not' ((
a 'imp' c) '&' (b 'imp' c)).x 'or' ((a 'or' b) 'imp' c).x by BVFUNC_1:def 8
.='not'( (a 'imp' c).x '&' (b 'imp' c).x) 'or' ((a 'or' b) 'imp' c).x
by MARGREL1:def 20
.='not'( ('not' a.x 'or' c.x) '&' (b 'imp' c).x) 'or' ((a 'or' b)
'imp' c).x by BVFUNC_1:def 8
.='not'( ('not' a.x 'or' c.x) '&' ('not' b.x 'or' c.x)) 'or' (
(a 'or' b) 'imp' c).x by BVFUNC_1:def 8
.='not'( ('not' a.x 'or' c.x) '&' ('not' b.x 'or' c.x)) 'or' (
'not' (a 'or' b).x 'or' c.x) by BVFUNC_1:def 8
.=('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x))
'or' (c.x 'or' 'not'( a.x 'or' b.x)) by BVFUNC_1:def 4
.=('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x))
'or' (('not' a.x 'or' c.x) '&' (c.x 'or' 'not' b.x)) by XBOOLEAN:9
.=(('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x)
) 'or' ('not' a.x 'or' c.x)) '&' (('not'( 'not' a.x 'or' c.x) 'or'
'not'( 'not' b.x 'or' c.x)) 'or' ('not' b.x 'or' c.x)) by XBOOLEAN:9
.=('not'( 'not' b.x 'or' c.x) 'or' ('not'( 'not' a.x 'or' c.x)
'or' ('not' a.x 'or' c.x))) '&' ('not'( 'not' a.x 'or' c.x) 'or' ('not'
( 'not' b.x 'or' c.x) 'or' ('not' b.x 'or' c.x)))
.=('not'( 'not' b.x 'or' c.x) 'or' TRUE) '&' ('not'( 'not' a.x
'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ('not' b.x 'or' c.x))
) by XBOOLEAN:102
.=TRUE '&' ('not'( 'not' a.x 'or' c.x) 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'imp' (b '&' 'not'
b)) 'imp' 'not' a=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'imp' (b '&' 'not' b)) 'imp' 'not' a)
.x=TRUE
proof
let x be Element of Y;
((a 'imp' (b '&' 'not' b)) 'imp' 'not' a).x ='not' (a 'imp' (b '&'
'not' b)).x 'or' ('not' a).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' (b '&' 'not' b).x) 'or' ('not' a).x by
BVFUNC_1:def 8
.=(a.x '&' ('not' b.x 'or' 'not' ('not' b).x)) 'or' ('not' a).x by
MARGREL1:def 20
.=(a.x '&' ('not' b.x 'or' 'not' 'not' b.x)) 'or' ('not' a).x by
MARGREL1:def 19
.=(a.x '&' TRUE) 'or' ('not' a).x by XBOOLEAN:102
.=a.x 'or' 'not' a.x by MARGREL1:def 19
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) '&' (a
'or' c)) 'imp' (a 'or' (b '&' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'or' b) '&' (a 'or' c)) 'imp' (a
'or' (b '&' c))).x=TRUE
proof
let x be Element of Y;
(((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))).x ='not' ((a
'or' b) '&' (a 'or' c)).x 'or' (a 'or' (b '&' c)).x by BVFUNC_1:def 8
.='not'( (a 'or' b).x '&' (a 'or' c).x) 'or' (a 'or' (b '&' c)).x by
MARGREL1:def 20
.='not' ((a.x 'or' b.x) '&' (a 'or' c).x) 'or' (a 'or' (b '&' c)).
x by BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) '&' (a.x 'or' c.x)) 'or' (a 'or' (b
'&' c)).x by BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) '&' (a.x 'or' c.x)) 'or' (a.x 'or' (
b '&' c).x) by BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) '&' (a.x 'or' c.x)) 'or' (a.x 'or' (
b.x '&' c.x)) by MARGREL1:def 20
.=('not'( a.x 'or' b.x) 'or' 'not'( a.x 'or' c.x)) 'or' ((a.
x 'or' b.x) '&' (a.x 'or' c.x)) by XBOOLEAN:9
.=(('not'( a.x 'or' c.x) 'or' 'not'( a.x 'or' b.x)) 'or' (a.
x 'or' b.x)) '&' (('not'( a.x 'or' b.x) 'or' 'not'( a.x 'or' c.x))
'or' (a.x 'or' c.x)) by XBOOLEAN:9
.=('not'( a.x 'or' c.x) 'or' ('not'( a.x 'or' b.x) 'or' (a.x
'or' b.x))) '&' ('not'( a.x 'or' b.x) 'or' ('not'( a.x 'or' c.x) 'or'
(a.x 'or' c.x)))
.=('not'( a.x 'or' c.x) 'or' TRUE) '&' ('not'( a.x 'or' b.x)
'or' ('not'( a.x 'or' c.x) 'or' (a.x 'or' c.x))) by XBOOLEAN:102
.=TRUE '&' ('not'( a.x 'or' b.x) 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a '&' (b 'or' c))
'imp' ((a '&' b) 'or' (a '&' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds ((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or'
(a '&' c))).x=TRUE
proof
let x be Element of Y;
((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))).x ='not' (a '&'
(b 'or' c)).x 'or' ((a '&' b) 'or' (a '&' c)).x by BVFUNC_1:def 8
.='not'( a.x '&' (b 'or' c).x) 'or' ((a '&' b) 'or' (a '&' c)).x by
MARGREL1:def 20
.='not' (a.x '&' (b.x 'or' c.x)) 'or' ((a '&' b) 'or' (a '&' c))
.x by BVFUNC_1:def 4
.='not'( a.x '&' (b.x 'or' c.x)) 'or' ((a '&' b).x 'or' (a '&' c
).x) by BVFUNC_1:def 4
.='not'( a.x '&' (b.x 'or' c.x)) 'or' ((a.x '&' b.x) 'or' (a
'&' c).x) by MARGREL1:def 20
.='not'( a.x '&' (b.x 'or' c.x)) 'or' ((a.x '&' b.x) 'or' ((
a).x '&' c.x)) by MARGREL1:def 20
.= ((a.x '&' b.x) 'or' (a.x '&' c.x)) 'or' ('not'( a.x '&' (
b).x) '&' 'not' (a.x '&' c.x)) by XBOOLEAN:8
.= (((a.x '&' c.x) 'or' (a.x '&' b.x)) 'or' 'not'( a.x '&' (
b).x)) '&' (((a.x '&' b.x) 'or' (a.x '&' c.x)) 'or' 'not'( a.x '&' (c
).x)) by XBOOLEAN:9
.= ((a.x '&' c.x) 'or' ((a.x '&' b.x) 'or' 'not'( a.x '&' (b
).x))) '&' ((a.x '&' b.x) 'or' ((a.x '&' c.x) 'or' 'not'( a.x '&' (c)
.x)))
.= ((a.x '&' c.x) 'or' TRUE) '&' ((a.x '&' b.x) 'or' ((a.x
'&' c.x) 'or' 'not'( a.x '&' c.x))) by XBOOLEAN:102
.= TRUE '&' ((a.x '&' b.x) 'or' TRUE) by XBOOLEAN:102
.= TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a 'or' c) '&' (b
'or' c)) 'imp' ((a '&' b) 'or' c)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'or' c) '&' (b 'or' c)) 'imp' ((a
'&' b) 'or' c)).x=TRUE
proof
let x be Element of Y;
(((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)).x ='not' ((a
'or' c) '&' (b 'or' c)).x 'or' ((a '&' b) 'or' c).x by BVFUNC_1:def 8
.='not'( (a 'or' c).x '&' (b 'or' c).x) 'or' ((a '&' b) 'or' c).x by
MARGREL1:def 20
.='not' ((a.x 'or' c.x) '&' (b 'or' c).x) 'or' ((a '&' b) 'or' c).
x by BVFUNC_1:def 4
.=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' ((a
'&' b) 'or' c).x by BVFUNC_1:def 4
.=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' ((a
'&' b).x 'or' c.x) by BVFUNC_1:def 4
.=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' (c.x
'or' (a.x '&' b.x)) by MARGREL1:def 20
.=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' ((a.
x 'or' c.x) '&' (c.x 'or' b.x)) by XBOOLEAN:9
.=(('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' (a.
x 'or' c.x)) '&' (('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x))
'or' (b.x 'or' c.x)) by XBOOLEAN:9
.=('not'( b.x 'or' c.x) 'or' ('not'( a.x 'or' c.x) 'or' (a.x
'or' c.x))) '&' ('not'( a.x 'or' c.x) 'or' ('not'( b.x 'or' c.x) 'or'
(b.x 'or' c.x)))
.=('not'( b.x 'or' c.x) 'or' TRUE) '&' ('not'( a.x 'or' c.x)
'or' ('not'( b.x 'or' c.x) 'or' (b.x 'or' c.x))) by XBOOLEAN:102
.=TRUE '&' ('not'( a.x 'or' c.x) 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) '&' c)
'imp' ((a '&' c) 'or' (b '&' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or'
(b '&' c))).x=TRUE
proof
let x be Element of Y;
(((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))).x ='not' ((a
'or' b) '&' c).x 'or' ((a '&' c) 'or' (b '&' c)).x by BVFUNC_1:def 8
.='not'( ((a 'or' b)).x '&' c.x) 'or' ((a '&' c) 'or' (b '&' c)).x
by MARGREL1:def 20
.='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a '&' c) 'or' (b '&' c))
.x by BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a '&' c).x 'or' (b '&' c
).x) by BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a.x '&' c.x) 'or' (b
'&' c).x) by MARGREL1:def 20
.='not'( c.x '&' (a.x 'or' b.x)) 'or' ((a.x '&' c.x) 'or' ((
b).x '&' c.x)) by MARGREL1:def 20
.=((a.x '&' c.x) 'or' (b.x '&' c.x)) 'or' ('not'( a.x '&' (c
).x) '&' 'not'( b.x '&' c.x)) by XBOOLEAN:8
.=(((b.x '&' c.x) 'or' (a.x '&' c.x)) 'or' 'not'( a.x '&' (c
).x)) '&' (((a.x '&' c.x) 'or' (b.x '&' c.x)) 'or' 'not'( b.x '&' (c)
.x)) by XBOOLEAN:9
.=((b.x '&' c.x) 'or' ((a.x '&' c.x) 'or' 'not'( a.x '&' (c)
.x))) '&' ((a.x '&' c.x) 'or' ((b.x '&' c.x) 'or' 'not'( b.x '&' c.
x)))
.=((b.x '&' c.x) 'or' TRUE) '&' ((a.x '&' c.x) 'or' ((b.x
'&' c.x) 'or' 'not'( b.x '&' c.x))) by XBOOLEAN:102
.=TRUE '&' ((a.x '&' c.x) 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a '&' b)=I_el(Y)
implies (a 'or' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
assume
A1: (a '&' b)=I_el(Y);
for x being Element of Y holds (a 'or' b).x=TRUE
proof
let x be Element of Y;
(a '&' b).x= TRUE by A1,BVFUNC_1:def 11;
then
A2: a.x '&' b.x=TRUE by MARGREL1:def 20;
then a.x=TRUE by MARGREL1:12;
then (a 'or' b).x =TRUE 'or' TRUE by A2,BVFUNC_1:def 4
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y)
implies (a 'or' c) 'imp' (b 'or' c)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
assume
A1: (a 'imp' b)=I_el(Y);
for x being Element of Y holds ((a 'or' c) 'imp' (b 'or' c)).x=TRUE
proof
let x be Element of Y;
(a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11;
then
A2: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8;
((a 'or' c) 'imp' (b 'or' c)).x ='not' (a 'or' c).x 'or' (b 'or' c).x
by BVFUNC_1:def 8
.='not'( a.x 'or' c.x) 'or' (b 'or' c).x by BVFUNC_1:def 4
.=(b.x 'or' c.x) 'or' ('not' a.x '&' 'not' c.x) by BVFUNC_1:def 4
.=((c.x 'or' b.x) 'or' 'not' a.x) '&' ((b.x 'or' c.x) 'or'
'not' c.x) by XBOOLEAN:9
.=(c.x 'or' ('not' a.x 'or' b.x)) '&' ((b.x 'or' c.x) 'or'
'not' c.x)
.=TRUE '&' (b.x 'or' (c.x 'or' 'not' c.x)) by A2
.=TRUE '&' (b.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y)
implies (a '&' c) 'imp' (b '&' c)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
assume
A1: (a 'imp' b)=I_el(Y);
for x being Element of Y holds ((a '&' c) 'imp' (b '&' c)).x=TRUE
proof
let x be Element of Y;
(a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11;
then
A2: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8;
((a '&' c) 'imp' (b '&' c)).x ='not' (a '&' c).x 'or' (b '&' c).x by
BVFUNC_1:def 8
.='not'( a.x '&' c.x) 'or' (b '&' c).x by MARGREL1:def 20
.=('not' a.x 'or' 'not' c.x) 'or' (b.x '&' c.x) by
MARGREL1:def 20
.=(('not' c.x 'or' 'not' a.x) 'or' b.x) '&' (('not' a.x 'or'
'not' c.x) 'or' c.x) by XBOOLEAN:9
.=('not' c.x 'or' ('not' a.x 'or' b.x)) '&' ('not' a.x 'or' (
'not' c.x 'or' c.x))
.=('not' c.x 'or' ('not' a.x 'or' b.x)) '&' ('not' a.x 'or'
TRUE) by XBOOLEAN:102
.=TRUE by A2;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem th18:
for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a)=I_el(Y)
& (c 'imp' b)=I_el(Y) implies c 'imp' (a '&' b)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
assume that
A1: (c 'imp' a)=I_el(Y) and
A2: (c 'imp' b)=I_el(Y);
for x being Element of Y holds (c 'imp' (a '&' b)).x=TRUE
proof
let x be Element of Y;
(c 'imp' a).x= TRUE by A1,BVFUNC_1:def 11;
then
A3: 'not' c.x 'or' a.x = TRUE by BVFUNC_1:def 8;
(c 'imp' b).x= TRUE by A2,BVFUNC_1:def 11;
then
A4: 'not' c.x 'or' b.x = TRUE by BVFUNC_1:def 8;
(c 'imp' (a '&' b)).x ='not' c.x 'or' (a '&' b).x by BVFUNC_1:def 8
.='not' c.x 'or' (a.x '&' b.x) by MARGREL1:def 20
.=TRUE '&' TRUE by A3,A4,XBOOLEAN:9
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c)=I_el(Y)
& (b 'imp' c)=I_el(Y) implies (a 'or' b) 'imp' c = I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
assume that
A1: (a 'imp' c)=I_el(Y) and
A2: (b 'imp' c)=I_el(Y);
for x being Element of Y holds ((a 'or' b) 'imp' c).x=TRUE
proof
let x be Element of Y;
(a 'imp' c).x= TRUE by A1,BVFUNC_1:def 11;
then
A3: 'not' a.x 'or' c.x = TRUE by BVFUNC_1:def 8;
(b 'imp' c).x= TRUE by A2,BVFUNC_1:def 11;
then
A4: 'not' b.x 'or' c.x = TRUE by BVFUNC_1:def 8;
((a 'or' b) 'imp' c).x ='not' (a 'or' b).x 'or' c.x by BVFUNC_1:def 8
.='not'( a.x 'or' b.x) 'or' c.x by BVFUNC_1:def 4
.=TRUE '&' TRUE by A3,A4,XBOOLEAN:9
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'or' b)=I_el(Y) &
'not' a=I_el(Y) implies b=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
assume that
A1: (a 'or' b)=I_el(Y) and
A2: 'not' a=I_el(Y);
for x being Element of Y holds b.x=TRUE
proof
let x be Element of Y;
('not' a).x= TRUE by A2,BVFUNC_1:def 11;
then
A3: 'not' a.x = TRUE by MARGREL1:def 19;
(a 'or' b).x= TRUE by A1,BVFUNC_1:def 11;
then a.x 'or' b.x = TRUE by BVFUNC_1:def 4;
hence thesis by A3;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem tt:
for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y
) & (c 'imp' d)=I_el(Y) implies (a '&' c) 'imp' (b '&' d)=I_el(Y)
proof
let a,b,c,d be Function of Y,BOOLEAN;
assume that
A1: (a 'imp' b)=I_el(Y) and
A2: (c 'imp' d)=I_el(Y);
for x being Element of Y holds ((a '&' c) 'imp' (b '&' d)).x=TRUE
proof
let x be Element of Y;
(a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11;
then
A3: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8;
(c 'imp' d).x= TRUE by A2,BVFUNC_1:def 11;
then
A4: 'not' c.x 'or' (d).x = TRUE by BVFUNC_1:def 8;
((a '&' c) 'imp' (b '&' d)).x ='not' (a '&' c).x 'or' (b '&' d).x by
BVFUNC_1:def 8
.='not'( a.x '&' c.x) 'or' (b '&' d).x by MARGREL1:def 20
.=('not' a.x 'or' 'not' c.x) 'or' (b.x '&' (d).x) by
MARGREL1:def 20
.=(('not' c.x 'or' 'not' a.x) 'or' b.x) '&' (('not' a.x 'or'
'not' c.x) 'or' (d).x) by XBOOLEAN:9
.=('not' c.x 'or' ('not' a.x 'or' b.x)) '&' (('not' a.x 'or'
'not' c.x) 'or' (d).x)
.=TRUE '&' ('not' a.x 'or' TRUE) by A3,A4,BINARITH:11
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem Th22:
for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y
) & (c 'imp' d)=I_el(Y) implies (a 'or' c) 'imp' (b 'or' d) =I_el(Y)
proof
let a,b,c,d be Function of Y,BOOLEAN;
assume that
A1: (a 'imp' b)=I_el(Y) and
A2: (c 'imp' d)=I_el(Y);
for x being Element of Y holds ((a 'or' c) 'imp' (b 'or' d)).x=TRUE
proof
let x be Element of Y;
(a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11;
then
A3: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8;
(c 'imp' d).x= TRUE by A2,BVFUNC_1:def 11;
then
A4: 'not' c.x 'or' (d).x = TRUE by BVFUNC_1:def 8;
((a 'or' c) 'imp' (b 'or' d)).x ='not' (a 'or' c).x 'or' (b 'or' d).x
by BVFUNC_1:def 8
.='not'( a.x 'or' c.x) 'or' (b 'or' d).x by BVFUNC_1:def 4
.=(b.x 'or' (d).x) 'or' ('not' a.x '&' 'not' c.x) by BVFUNC_1:def 4
.=(((d).x 'or' b.x) 'or' 'not' a.x) '&' ((b.x 'or' (d).x) 'or'
'not' c.x) by XBOOLEAN:9
.=((d).x 'or' (b.x 'or' 'not' a.x)) '&' ((b.x 'or' (d).x) 'or'
'not' c.x)
.=TRUE '&' (b.x 'or' TRUE) by A3,A4,BINARITH:11
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a '&' 'not' b) 'imp'
'not' a=I_el(Y) implies (a 'imp' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
assume
A1: (a '&' 'not' b) 'imp' 'not' a=I_el(Y);
for x being Element of Y holds (a 'imp' b).x=TRUE
proof
let x be Element of Y;
((a '&' 'not' b) 'imp' 'not' a).x=TRUE by A1,BVFUNC_1:def 11;
then 'not' (a '&' 'not' b).x 'or' ('not' a).x = TRUE by BVFUNC_1:def 8;
then 'not'( a.x '&' ('not' b).x) 'or' ('not' a).x=TRUE by MARGREL1:def 20
;
then ('not' a.x 'or' 'not' 'not' b.x) 'or' ('not' a).x=TRUE by
MARGREL1:def 19;
then ('not' a.x 'or' b.x) 'or' 'not' a.x=TRUE by MARGREL1:def 19;
then b.x 'or' ('not' a.x 'or' 'not' a.x)=TRUE by XBOOLEAN:4;
hence thesis by BVFUNC_1:def 8;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a 'imp' 'not' b=I_el(Y
) implies b 'imp' 'not' a=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
assume
A1: a 'imp' 'not' b=I_el(Y);
for x being Element of Y holds (b 'imp' 'not' a).x=TRUE
proof
let x be Element of Y;
(a 'imp' 'not' b).x=TRUE by A1,BVFUNC_1:def 11;
then ('not' a.x) 'or' ('not' b).x=TRUE by BVFUNC_1:def 8;
then
A2: 'not' a.x 'or' 'not' b.x=TRUE by MARGREL1:def 19;
(b 'imp' 'not' a).x =('not' b.x) 'or' ('not' a).x by BVFUNC_1:def 8
.=TRUE by A2,MARGREL1:def 19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not' a 'imp' b=I_el(Y
) implies 'not' b 'imp' a=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
assume
A1: 'not' a 'imp' b=I_el(Y);
for x being Element of Y holds ('not' b 'imp' a).x=TRUE
proof
let x be Element of Y;
('not' a 'imp' b).x=TRUE by A1,BVFUNC_1:def 11;
then ('not' ('not' a).x) 'or' b.x=TRUE by BVFUNC_1:def 8;
then
A2: 'not' 'not' a.x 'or' b.x=TRUE by MARGREL1:def 19;
('not' b 'imp' a).x =('not' ('not' b).x) 'or' a.x by BVFUNC_1:def 8
.=TRUE by A2,MARGREL1:def 19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem Th26:
for a,b being Function of Y,BOOLEAN holds a 'imp' (a 'or' b)= I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds (a 'imp' (a 'or' b)).x=TRUE
proof
let x be Element of Y;
(a 'imp' (a 'or' b)).x ='not' a.x 'or' (a 'or' b).x by BVFUNC_1:def 8
.='not' a.x 'or' (a.x 'or' b.x) by BVFUNC_1:def 4
.=('not' a.x 'or' a.x) 'or' b.x
.=TRUE 'or' b.x by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'or' b) 'imp' (
'not' a 'imp' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'or' b) 'imp' ('not' a 'imp' b)).x= TRUE
proof
let x be Element of Y;
((a 'or' b) 'imp' ('not' a 'imp' b)).x ='not' (a 'or' b).x 'or' ('not'
a 'imp' b).x by BVFUNC_1:def 8
.='not'( a.x 'or' b.x) 'or' ('not' a 'imp' b).x by BVFUNC_1:def 4
.=('not' a.x '&' 'not' b.x) 'or' ('not' ('not' a).x 'or' b.x) by
BVFUNC_1:def 8
.=(a.x 'or' b.x) 'or' ('not' a.x '&' 'not' b.x) by
MARGREL1:def 19
.=((a.x 'or' b.x) 'or' 'not' a.x) '&' ((a.x 'or' b.x) 'or'
'not' b.x) by XBOOLEAN:9
.=((a.x 'or' 'not' a.x) 'or' b.x) '&' (a.x 'or' (b.x 'or'
'not' b.x))
.=(TRUE 'or' b.x) '&' (a.x 'or' (b.x 'or' 'not' b.x)) by
XBOOLEAN:102
.=TRUE '&' (a.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem Th28:
for a,b being Function of Y,BOOLEAN holds 'not'( a 'or' b)
'imp' ('not' a '&' 'not' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ('not'( a 'or' b) 'imp' ('not' a '&'
'not' b)).x=TRUE
proof
let x be Element of Y;
('not'( a 'or' b) 'imp' ('not' a '&' 'not' b)).x ='not' ('not'( a 'or'
b)).x 'or' ('not' a '&' 'not' b).x by BVFUNC_1:def 8
.=(a 'or' b).x 'or' ('not' a '&' 'not' b).x by MARGREL1:def 19
.=(a.x 'or' b.x) 'or' ('not' a '&' 'not' b).x by BVFUNC_1:def 4
.=(a.x 'or' b.x) 'or' (('not' a).x '&' ('not' b).x) by
MARGREL1:def 20
.=(a.x 'or' b.x) 'or' ('not' a.x '&' ('not' b).x) by
MARGREL1:def 19
.=(a.x 'or' b.x) 'or' ('not' a.x '&' 'not' b.x) by
MARGREL1:def 19
.=((a.x 'or' b.x) 'or' 'not' a.x) '&' ((a.x 'or' b.x) 'or'
'not' b.x) by XBOOLEAN:9
.=((a.x 'or' 'not' a.x) 'or' b.x) '&' (a.x 'or' (b.x 'or'
'not' b.x))
.=(TRUE 'or' b.x) '&' (a.x 'or' (b.x 'or' 'not' b.x)) by
XBOOLEAN:102
.=TRUE '&' (a.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds ('not' a '&' 'not' b)
'imp' 'not'( a 'or' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
thus ('not' a '&' 'not' b) 'imp' 'not'( a 'or' b) = 'not'( a 'or' b) 'imp'
'not' (a 'or' b) by BVFUNC_1:13
.= 'not'( a 'or' b) 'imp' ('not' a '&' 'not' b) by BVFUNC_1:13
.=I_el Y by Th28;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not'( a 'or' b) 'imp'
'not' a=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ('not'( a 'or' b) 'imp' 'not' a).x=TRUE
proof
let x be Element of Y;
('not'( a 'or' b) 'imp' 'not' a).x ='not' ('not'( a 'or' b)).x 'or' (
'not' a).x by BVFUNC_1:def 8
.='not' 'not' (a 'or' b).x 'or' ('not' a).x by MARGREL1:def 19
.=(a 'or' b).x 'or' 'not' a.x by MARGREL1:def 19
.=(a.x 'or' b.x) 'or' 'not' a.x by BVFUNC_1:def 4
.=(a.x 'or' 'not' a.x) 'or' b.x
.=TRUE 'or' b.x by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a being Function of Y,BOOLEAN holds (a 'or' a) 'imp' a=I_el( Y)
proof
let a be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'or' a) 'imp' a).x=TRUE
proof
let x be Element of Y;
((a 'or' a) 'imp' a).x ='not' a.x 'or' a.x by BVFUNC_1:def 8
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a '&' 'not' a) 'imp'
b=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a '&' 'not' a) 'imp' b).x=TRUE
proof
let x be Element of Y;
((a '&' 'not' a) 'imp' b).x ='not' (a '&' 'not' a).x 'or' b.x by
BVFUNC_1:def 8
.='not'( a.x '&' ('not' a).x) 'or' b.x by MARGREL1:def 20
.=('not' a.x 'or' 'not' 'not' a.x) 'or' b.x by MARGREL1:def 19
.=TRUE 'or' b.x by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (
'not' a 'or' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'imp' b) 'imp' ('not' a 'or' b)).x= TRUE
proof
let x be Element of Y;
((a 'imp' b) 'imp' ('not' a 'or' b)).x ='not' (a 'imp' b).x 'or' (
'not' a 'or' b).x by BVFUNC_1:def 8
.='not'( 'not' a.x 'or' b.x) 'or' ('not' a 'or' b).x by
BVFUNC_1:def 8
.=('not' 'not' a.x '&' 'not' b.x) 'or' (('not' a).x 'or' b.x) by
BVFUNC_1:def 4
.=('not' a.x 'or' b.x) 'or' (a.x '&' 'not' b.x) by
MARGREL1:def 19
.=(('not' a.x 'or' b.x) 'or' a.x) '&' (('not' a.x 'or' b.x)
'or' 'not' b.x) by XBOOLEAN:9
.=(('not' a.x 'or' a.x) 'or' b.x) '&' ('not' a.x 'or' (b.x
'or' 'not' b.x))
.=(TRUE 'or' b.x) '&' ('not' a.x 'or' (b.x 'or' 'not' b.x)) by
XBOOLEAN:102
.=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' 'not'(
a 'imp' 'not' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a '&' b) 'imp' 'not'( a 'imp' 'not' b))
.x=TRUE
proof
let x be Element of Y;
((a '&' b) 'imp' 'not'( a 'imp' 'not' b)).x ='not' (a '&' b).x 'or' (
'not'( a 'imp' 'not' b)).x by BVFUNC_1:def 8
.='not'( a.x '&' b.x) 'or' ('not'( a 'imp' 'not' b)).x by
MARGREL1:def 20
.=('not' a.x 'or' 'not' b.x) 'or' 'not' (a 'imp' 'not' b).x by
MARGREL1:def 19
.=('not' a.x 'or' 'not' b.x) 'or' 'not'( 'not' a.x 'or' ('not' b
).x) by BVFUNC_1:def 8
.=('not' a.x 'or' 'not' b.x) 'or' (a.x '&' b.x) by
MARGREL1:def 19
.=(('not' a.x 'or' 'not' b.x) 'or' a.x) '&' (('not' a.x 'or'
'not' b.x) 'or' b.x) by XBOOLEAN:9
.=(('not' a.x 'or' a.x) 'or' 'not' b.x) '&' ('not' a.x 'or' (
'not' b.x 'or' b.x))
.=(TRUE 'or' 'not' b.x) '&' ('not' a.x 'or' ('not' b.x 'or' b.
x)) by XBOOLEAN:102
.=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not'( a 'imp' 'not' b
) 'imp' (a '&' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ('not'( a 'imp' 'not' b) 'imp' (a '&' b))
.x=TRUE
proof
let x be Element of Y;
('not'( a 'imp' 'not' b) 'imp' (a '&' b)).x ='not' ('not'( a 'imp'
'not' b)).x 'or' (a '&' b).x by BVFUNC_1:def 8
.='not' 'not' (a 'imp' 'not' b).x 'or' (a '&' b).x by MARGREL1:def 19
.='not' 'not' (a 'imp' 'not' b).x 'or' (a.x '&' b.x) by
MARGREL1:def 20
.=('not' a.x 'or' ('not' b).x) 'or' (a.x '&' b.x) by BVFUNC_1:def 8
.=('not' a.x 'or' 'not' b.x) 'or' (a.x '&' b.x) by
MARGREL1:def 19
.=(('not' a.x 'or' 'not' b.x) 'or' a.x) '&' (('not' a.x 'or'
'not' b.x) 'or' b.x) by XBOOLEAN:9
.=(('not' a.x 'or' a.x) 'or' 'not' b.x) '&' ('not' a.x 'or' (
'not' b.x 'or' b.x))
.=(TRUE 'or' 'not' b.x) '&' ('not' a.x 'or' ('not' b.x 'or' b.
x)) by XBOOLEAN:102
.=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem Th36:
for a,b being Function of Y,BOOLEAN holds 'not'( a '&' b)
'imp' ('not' a 'or' 'not' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ('not'( a '&' b) 'imp' ('not' a 'or'
'not' b)).x=TRUE
proof
let x be Element of Y;
('not'( a '&' b) 'imp' ('not' a 'or' 'not' b)).x ='not' ('not'( a '&'
b)).x 'or' ('not' a 'or' 'not' b).x by BVFUNC_1:def 8
.=(a '&' b).x 'or' ('not' a 'or' 'not' b).x by MARGREL1:def 19
.=(a.x '&' b.x) 'or' ('not' a 'or' 'not' b).x by MARGREL1:def 20
.=(('not' a).x 'or' ('not' b).x) 'or' (a.x '&' b.x) by BVFUNC_1:def 4
.=('not' a.x 'or' ('not' b).x) 'or' (a.x '&' b.x) by
MARGREL1:def 19
.=('not' a.x 'or' 'not' b.x) 'or' (a.x '&' b.x) by
MARGREL1:def 19
.=(('not' a.x 'or' 'not' b.x) 'or' a.x) '&' (('not' a.x 'or'
'not' b.x) 'or' b.x) by XBOOLEAN:9
.=(('not' a.x 'or' a.x) 'or' 'not' b.x) '&' ('not' a.x 'or' (
'not' b.x 'or' b.x))
.=(TRUE 'or' 'not' b.x) '&' ('not' a.x 'or' ('not' b.x 'or' b.
x)) by XBOOLEAN:102
.=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds ('not' a 'or' 'not' b)
'imp' 'not'( a '&' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
thus ('not' a 'or' 'not' b) 'imp' 'not'( a '&' b) = 'not'( a '&' b) 'imp'
'not' (a '&' b) by BVFUNC_1:14
.= 'not'( a '&' b) 'imp' ('not' a 'or' 'not' b) by BVFUNC_1:14
.=I_el Y by Th36;
end;
theorem Th38:
for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' a=I_el (Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a '&' b) 'imp' a).x=TRUE
proof
let x be Element of Y;
((a '&' b) 'imp' a).x ='not' (a '&' b).x 'or' a.x by BVFUNC_1:def 8
.=('not' a.x 'or' 'not' b.x) 'or' a.x by MARGREL1:def 20
.=('not' a.x 'or' a.x) 'or' 'not' b.x
.=TRUE 'or' 'not' b.x by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' (a
'or' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a '&' b) 'imp' (a 'or' b)).x=TRUE
proof
let x be Element of Y;
((a '&' b) 'imp' (a 'or' b)).x ='not' (a '&' b).x 'or' (a 'or' b).x by
BVFUNC_1:def 8
.=('not' a.x 'or' 'not' b.x) 'or' (a 'or' b).x by MARGREL1:def 20
.=('not' a.x 'or' 'not' b.x) 'or' (a.x 'or' b.x) by
BVFUNC_1:def 4
.=('not' b.x 'or' ('not' a.x 'or' a.x)) 'or' b.x
.=('not' b.x 'or' TRUE) 'or' b.x by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' b=I_el (Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a '&' b) 'imp' b).x=TRUE
proof
let x be Element of Y;
((a '&' b) 'imp' b).x ='not' (a '&' b).x 'or' b.x by BVFUNC_1:def 8
.=('not' a.x 'or' 'not' b.x) 'or' b.x by MARGREL1:def 20
.='not' a.x 'or' ('not' b.x 'or' b.x)
.='not' a.x 'or' TRUE by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'imp' a '&' a=I_el(Y)
proof
let a be Function of Y,BOOLEAN;
for x being Element of Y holds (a 'imp' (a '&' a)).x=TRUE
proof
let x be Element of Y;
(a 'imp' a '&' a).x =TRUE '&' ('not' a.x 'or' a.x) by BVFUNC_1:def 8
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem Th42:
for a,b being Function of Y,BOOLEAN holds (a 'eqv' b)
'imp' (a 'imp' b)=I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'eqv' b) 'imp' (a 'imp' b)).x=TRUE
proof
let x be Element of Y;
((a 'eqv' b) 'imp' (a 'imp' b)).x ='not' (a 'eqv' b).x 'or' (a 'imp' b
).x by BVFUNC_1:def 8
.=(a.x 'xor' b.x) 'or' (a 'imp' b).x by BVFUNC_1:def 9
.=(('not' a.x '&' b.x) 'or' (a.x '&' 'not' b.x)) 'or' 'not'( (
a).x '&' 'not' b.x) by BVFUNC_1:def 8
.=('not' a.x '&' b.x) 'or' ((a.x '&' 'not' b.x) 'or' 'not'( (a
).x '&' 'not' b.x))
.=('not' a.x '&' b.x) 'or' TRUE by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'eqv' b) 'imp' (b
'imp' a)=I_el(Y) by Th42;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c)
'imp' (a 'or' (b 'or' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'or' b) 'or' c) 'imp' (a 'or' (b
'or' c))).x=TRUE
proof
let x be Element of Y;
(((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))).x ='not' ((a 'or' b)
'or' c).x 'or' (a 'or' (b 'or' c)).x by BVFUNC_1:def 8
.='not'( (a 'or' b).x 'or' c.x) 'or' (a 'or' (b 'or' c)).x by
BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a 'or' (b 'or' c)).x by
BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a.x 'or' (b 'or' c).x)
by BVFUNC_1:def 4
.='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a.x 'or' (b.x 'or' (
c).x)) by BVFUNC_1:def 4
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) '&' c)
'imp' (a '&' (b '&' c))=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds (((a '&' b) '&' c) 'imp' (a '&' (b '&' c)
)).x=TRUE
proof
let x be Element of Y;
(((a '&' b) '&' c) 'imp' (a '&' (b '&' c))).x ='not' ((a '&' b) '&' c)
.x 'or' (a '&' (b '&' c)).x by BVFUNC_1:def 8
.='not'( (a '&' b).x '&' c.x) 'or' (a '&' (b '&' c)).x by
MARGREL1:def 20
.='not'( (a.x '&' b.x) '&' c.x) 'or' (a '&' (b '&' c)).x by
MARGREL1:def 20
.='not'( (a.x '&' b.x) '&' c.x) 'or' (a.x '&' (b '&' c).x) by
MARGREL1:def 20
.='not'( (a.x '&' b.x) '&' c.x) 'or' (a.x '&' (b.x '&' c.x
)) by MARGREL1:def 20
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'or' (b 'or' c))
'imp' ((a 'or' b) 'or' c)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'or' (b 'or' c)) 'imp' ((a 'or' b)
'or' c)).x=TRUE
proof
let x be Element of Y;
((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)).x ='not' (a 'or' (b
'or' c)).x 'or' ((a 'or' b) 'or' c).x by BVFUNC_1:def 8
.='not'( a.x 'or' (b 'or' c).x) 'or' ((a 'or' b) 'or' c).x by
BVFUNC_1:def 4
.='not'( a.x 'or' (b.x 'or' c.x)) 'or' ((a 'or' b) 'or' c).x by
BVFUNC_1:def 4
.='not'( a.x 'or' (b.x 'or' c.x)) 'or' ((a 'or' b).x 'or' c.x)
by BVFUNC_1:def 4
.='not'( a.x 'or' (b.x 'or' c.x)) 'or' ((a.x 'or' b.x) 'or'
c.x) by BVFUNC_1:def 4
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
begin :: BVFUNC_7
reserve Y for non empty set;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'imp' b) '&' ('not'
a 'imp' b) = b
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
A1: ((a 'imp' b) '&' ('not' a 'imp' b)).x =(a 'imp' b).x '&' ('not' a
'imp' b).x by MARGREL1:def 20
.=('not' a.x 'or' b.x) '&' ('not' a 'imp' b).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' ('not' a).x 'or' b.x) by
BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' (a.x 'or' b.x) by MARGREL1:def 19;
now
per cases by XBOOLEAN:def 3;
case
a.x=TRUE;
then ((a 'imp' b) '&' ('not' a 'imp' b)).x =(FALSE 'or' b.x) '&' (
TRUE 'or' b.x) by A1
.=(FALSE 'or' b.x) '&' TRUE
.=TRUE '&' b.x
.=b.x;
hence thesis;
end;
case
a.x=FALSE;
then ((a 'imp' b) '&' ('not' a 'imp' b)).x =(TRUE 'or' b.x) '&' (
FALSE 'or' b.x) by A1
.=TRUE '&' (FALSE 'or' b.x)
.=TRUE '&' b.x
.=b.x;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a
'imp' 'not' b) = 'not' a
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
A1: ((a 'imp' b) '&' (a 'imp' 'not' b)).x =(a 'imp' b).x '&' (a 'imp'
'not' b).x by MARGREL1:def 20
.=('not' a.x 'or' b.x) '&' (a 'imp' 'not' b).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' a.x 'or' ('not' b).x) by
BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' a.x 'or' 'not' b.x) by
MARGREL1:def 19;
now
per cases by XBOOLEAN:def 3;
case
b.x=TRUE;
then
((a 'imp' b) '&' (a 'imp' 'not' b)).x =('not' a.x 'or' TRUE) '&'
('not' a.x 'or' FALSE) by A1
.=('not' a.x 'or' TRUE) '&' 'not' a.x
.=TRUE '&' 'not' a.x
.='not' a.x
.=('not' a).x by MARGREL1:def 19;
hence thesis;
end;
case
b.x=FALSE;
then ((a 'imp' b) '&' (a 'imp' 'not' b)).x =('not' a.x 'or' FALSE)
'&' ('not' a.x 'or' TRUE) by A1
.='not' a.x '&' ('not' a.x 'or' TRUE)
.=TRUE '&' 'not' a.x
.='not' a.x
.=('not' a).x by MARGREL1:def 19;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th73:
for a,b,c being Function of Y,BOOLEAN holds a 'imp' (b 'or' c) =
(a 'imp' b) 'or' (a 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' b) 'or' (a 'imp' c)).x =(a 'imp' b).x 'or' (a 'imp' c).x by
BVFUNC_1:def 4
.=('not' a.x 'or' b.x) 'or' (a 'imp' c).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) 'or' ('not' a.x 'or' c.x) by
BVFUNC_1:def 8
.=('not' a.x 'or' ('not' a.x 'or' b.x)) 'or' c.x
.=(('not' a.x 'or' 'not' a.x) 'or' b.x) 'or' c.x by BINARITH:11
.='not' a.x 'or' (b.x 'or' c.x)
.='not' a.x 'or' (b 'or' c).x by BVFUNC_1:def 4
.=(a 'imp' (b 'or' c)).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds a 'imp' (b '&' c) =
(a 'imp' b) '&' (a 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' b) '&' (a 'imp' c)).x =(a 'imp' b).x '&' (a 'imp' c).x by
MARGREL1:def 20
.=('not' a.x 'or' b.x) '&' (a 'imp' c).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' a.x 'or' c.x) by BVFUNC_1:def 8
.='not' a.x 'or' (b.x '&' c.x) by XBOOLEAN:9
.='not' a.x 'or' ((b '&' c).x) by MARGREL1:def 20
.=(a 'imp' (b '&' c)).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem Th75:
for a,b,c being Function of Y,BOOLEAN holds (a 'or' b) 'imp' c =
(a 'imp' c) '&' (b 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' c) '&' (b 'imp' c)).x =(a 'imp' c).x '&' (b 'imp' c).x by
MARGREL1:def 20
.=('not' a.x 'or' c.x) '&' (b 'imp' c).x by BVFUNC_1:def 8
.=(c.x 'or' 'not' a.x) '&' ('not' b.x 'or' c.x) by BVFUNC_1:def 8
.='not'( a.x 'or' b.x) 'or' c.x by XBOOLEAN:9
.='not' (a 'or' b).x 'or' c.x by BVFUNC_1:def 4
.=((a 'or' b) 'imp' c).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem Th76:
for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c =
(a 'imp' c) 'or' (b 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' c) 'or' (b 'imp' c)).x =(a 'imp' c).x 'or' (b 'imp' c).x by
BVFUNC_1:def 4
.=('not' a.x 'or' c.x) 'or' (b 'imp' c).x by BVFUNC_1:def 8
.=('not' a.x 'or' c.x) 'or' ('not' b.x 'or' c.x) by
BVFUNC_1:def 8
.=('not' a.x 'or' (c.x 'or' 'not' b.x)) 'or' c.x
.=(('not' a.x 'or' 'not' b.x) 'or' c.x) 'or' c.x
.=('not' a.x 'or' 'not' b.x) 'or' (c.x 'or' c.x) by BINARITH:11
.='not' (a '&' b).x 'or' c.x by MARGREL1:def 20
.=((a '&' b) 'imp' c).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem Th7:
for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'imp'
c = a 'imp' (b 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'imp' (b 'imp' c)).x ='not' a.x 'or' (b 'imp' c).x by BVFUNC_1:def 8
.='not' a.x 'or' ('not' b.x 'or' c.x) by BVFUNC_1:def 8
.=('not' a.x 'or' 'not' b.x) 'or' c.x
.='not' (a '&' b).x 'or' c.x by MARGREL1:def 20
.=((a '&' b) 'imp' c).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c =
a 'imp' ('not' b 'or' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'imp' ('not' b 'or' c)).x =(a 'imp' (b 'imp' c)).x by BVFUNC_4:8;
hence thesis by Th7;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds a 'imp' (b 'or' c) =
(a '&' 'not' b) 'imp' c
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a '&' 'not' b) 'imp' c).x ='not' (a '&' 'not' b).x 'or' c.x by
BVFUNC_1:def 8
.=('not' a.x 'or' 'not' ('not' b).x) 'or' c.x by MARGREL1:def 20
.=('not' a.x 'or' b.x) 'or' c.x by MARGREL1:def 19
.='not' a.x 'or' (b.x 'or' c.x)
.='not' a.x 'or' (b 'or' c).x by BVFUNC_1:def 4
.=(a 'imp' (b 'or' c)).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '&' (a 'imp' b) = a '&' b
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
(a '&' (a 'imp' b)).x =a.x '&' (a 'imp' b).x by MARGREL1:def 20
.=a.x '&' ('not' a.x 'or' b.x) by BVFUNC_1:def 8
.=(a.x '&' 'not' a.x) 'or' (a.x '&' b.x) by XBOOLEAN:8
.=FALSE 'or' (a.x '&' b.x) by XBOOLEAN:138
.=a.x '&' b.x
.=(a '&' b).x by MARGREL1:def 20;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds (a 'imp' b) '&' 'not'
b = 'not' a '&' 'not' b
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' b) '&' 'not' b).x =(a 'imp' b).x '&' ('not' b).x by
MARGREL1:def 20
.=('not' b).x '&' ('not' a.x 'or' b.x) by BVFUNC_1:def 8
.=(('not' b).x '&' 'not' a.x) 'or' (('not' b).x '&' b.x) by
XBOOLEAN:8
.=(('not' b).x '&' 'not' a.x) 'or' (b.x '&' 'not' b.x) by
MARGREL1:def 19
.=(('not' b).x '&' 'not' a.x) 'or' FALSE by XBOOLEAN:138
.=('not' b).x '&' 'not' a.x
.=('not' b).x '&' ('not' a).x by MARGREL1:def 19
.=('not' a '&' 'not' b).x by MARGREL1:def 20;
hence thesis;
end;
theorem Th12:
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) = (a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =((a 'imp' b) '&' (b
'imp' c)).x '&' ((a 'imp' c)).x by MARGREL1:def 20
.=((a 'imp' b).x '&' (b 'imp' c).x) '&' ((a 'imp' c)).x by
MARGREL1:def 20
.=(('not' a.x 'or' b.x) '&' (b 'imp' c).x) '&' ((a 'imp' c)).x by
BVFUNC_1:def 8
.=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' ((a
'imp' c)).x by BVFUNC_1:def 8
.=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' ('not' (
a).x 'or' c.x) by BVFUNC_1:def 8
.=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' 'not' (a
).x 'or' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' c.x by
XBOOLEAN:8;
A2: (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) =((a 'imp' b).
x '&' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8
.=(a 'imp' b).x '&' (b 'imp' c).x by BVFUNC_1:def 8
.=((a 'imp' b) '&' (b 'imp' c)).x by MARGREL1:def 20;
A3: ((a 'imp' b) '&' (b 'imp' c)).x =(a 'imp' b).x '&' (b 'imp' c).x by
MARGREL1:def 20
.=('not' a.x 'or' b.x) '&' (b 'imp' c).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x) by BVFUNC_1:def 8
;
now
per cases by XBOOLEAN:def 3;
case
a.x=TRUE & c.x=TRUE;
then ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =(('not' a.x
'or' b.x) '&' ('not' b.x 'or' c.x)) '&' FALSE 'or' (('not' a.x 'or' (b)
.x) '&' ('not' b.x 'or' c.x)) '&' TRUE by A1
.=FALSE 'or' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x
)) '&' TRUE
.=FALSE 'or' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x
))
.=((a 'imp' b) '&' (b 'imp' c)).x by A2;
hence thesis;
end;
case
A4: a.x=TRUE & c.x=FALSE;
then
A5: ((a 'imp' b) '&' (b 'imp' c)).x =(FALSE 'or' b.x) '&' ('not' (b
).x 'or' FALSE) by A3
.=(FALSE 'or' b.x) '&' 'not' b.x
.=b.x '&' 'not' b.x
.=FALSE by XBOOLEAN:138;
((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =(('not' a.x
'or' b.x) '&' ('not' b.x 'or' c.x)) '&' FALSE 'or' (('not' a.x 'or' (b)
.x) '&' ('not' b.x 'or' c.x)) '&' FALSE by A1,A4
.=FALSE;
hence thesis by A5;
end;
case
a.x=FALSE & c.x=TRUE;
then ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =(('not' a.x
'or' b.x) '&' ('not' b.x 'or' c.x)) '&' TRUE 'or' (('not' a.x 'or' b.
x) '&' ('not' b.x 'or' c.x)) '&' TRUE by A1
.=((a 'imp' b) '&' (b 'imp' c)).x by A2;
hence thesis;
end;
case
a.x=FALSE & c.x=FALSE;
then
((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =TRUE '&' (('not'
a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) 'or' FALSE '&' (('not' a.x
'or' b.x) '&' ('not' b.x 'or' c.x)) by A1
.=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) 'or'
FALSE '&' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x))
.=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) 'or'
FALSE
.=((a 'imp' b) '&' (b 'imp' c)).x by A2;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds I_el(Y) 'imp' a = a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(I_el(Y) 'imp' a).x ='not' (I_el(Y)).x 'or' a.x by BVFUNC_1:def 8
.=FALSE 'or' a.x by BVFUNC_1:def 11;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'imp' O_el(Y) = 'not' a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'imp' O_el(Y)).x ='not' a.x 'or' (O_el(Y)).x by BVFUNC_1:def 8
.='not' a.x 'or' FALSE by BVFUNC_1:def 10
.='not' a.x;
hence thesis by MARGREL1:def 19;
end;
theorem
for a being Function of Y,BOOLEAN holds O_el(Y) 'imp' a = I_el(Y )
proof
let a be Function of Y,BOOLEAN;
for x being Element of Y holds (O_el(Y) 'imp' a).x = TRUE
proof
let x be Element of Y;
(O_el(Y) 'imp' a).x ='not' (O_el(Y)).x 'or' a.x by BVFUNC_1:def 8
.=TRUE 'or' a.x by BVFUNC_1:def 10;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'imp' I_el(Y) = I_el(Y )
proof
let a be Function of Y,BOOLEAN;
for x being Element of Y holds (a 'imp' I_el(Y)).x = TRUE
proof
let x be Element of Y;
(a 'imp' I_el(Y)).x ='not' a.x 'or' (I_el(Y)).x by BVFUNC_1:def 8
.='not' a.x 'or' TRUE by BVFUNC_1:def 11;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'imp' 'not' a = 'not' a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'imp' 'not' a).x ='not' a.x 'or' ('not' a).x by BVFUNC_1:def 8
.=('not' a).x 'or' ('not' a).x by MARGREL1:def 19
.=('not' a).x;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '<' (c
'imp' a) 'imp' (c 'imp' b)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
assume (a 'imp' b).z=TRUE;
then
A1: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8;
A2: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
now
per cases by A1,A2;
case
A3: 'not' a.z=TRUE;
((c 'imp' a) 'imp' (c 'imp' b)).z ='not' (c 'imp' a).z 'or' (c 'imp'
b).z by BVFUNC_1:def 8
.='not'( 'not' c.z 'or' a.z) 'or' (c 'imp' b).z by BVFUNC_1:def 8
.=c.z 'or' (c 'imp' b).z by A3
.=c.z 'or' ('not' c.z 'or' b.z) by BVFUNC_1:def 8
.=(c.z 'or' 'not' c.z) 'or' b.z
.=TRUE 'or' b.z by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
case
A4: b.z=TRUE;
((c 'imp' a) 'imp' (c 'imp' b)).z ='not' (c 'imp' a).z 'or' (c 'imp'
b).z by BVFUNC_1:def 8
.='not'( 'not' c.z 'or' a.z) 'or' (c 'imp' b).z by BVFUNC_1:def 8
.='not'( 'not' c.z 'or' a.z) 'or' ('not' c.z 'or' TRUE) by A4,
BVFUNC_1:def 8
.='not'( 'not' c.z 'or' a.z) 'or' TRUE
.=TRUE;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a
'eqv' c) 'eqv' (b 'eqv' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7
.=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20;
assume
A2: (a 'eqv' b).z=TRUE;
then (a 'imp' b).z=TRUE by A1,MARGREL1:12;
then
A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8;
A4: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
A5: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
A6: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8;
A7: ((a 'eqv' c) 'eqv' (b 'eqv' c)).z =(((a 'eqv' c) 'imp' (b 'eqv' c)) '&'
((b 'eqv' c) 'imp' (a 'eqv' c))).z by BVFUNC_4:7
.=((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'eqv' c)) '&' ((b 'eqv' c)
'imp' (a 'eqv' c))).z by BVFUNC_4:7
.=((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'imp' c) '&' (c 'imp' b)) '&'
((b 'eqv' c) 'imp' (a 'eqv' c))).z by BVFUNC_4:7
.=((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'imp' c) '&' (c 'imp' b)) '&'
(((b 'imp' c) '&' (c 'imp' b)) 'imp' (a 'eqv' c))).z by BVFUNC_4:7
.=((((a 'imp' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b)))
'&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z
by BVFUNC_4:7
.=(((('not' a 'or' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b
))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z
by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' ((b 'imp' c) '&' (c
'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a
)))).z by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&'
(c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c
'imp' a)))).z by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&'
('not' c 'or' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c
'imp' a)))).z by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&'
('not' c 'or' b))) '&' ((('not' b 'or' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c)
'&' (c 'imp' a)))).z by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&'
('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' ((a 'imp'
c) '&' (c 'imp' a)))).z by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&'
('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' (('not' a
'or' c) '&' (c 'imp' a)))).z by BVFUNC_4:8
.=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&'
('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' (('not' a
'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_4:8
.=(('not'( ('not' a 'or' c) '&' ('not' c 'or' a)) 'or' (('not' b 'or' c)
'&' ('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' ((
'not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_4:8
.=(('not'( ('not' a 'or' c) '&' ('not' c 'or' a)) 'or' (('not' b 'or' c)
'&' ('not' c 'or' b))) '&' ('not'( ('not' b 'or' c) '&' ('not' c 'or' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_4:8
.=((('not'( 'not' a 'or' c) 'or' 'not'( 'not' c 'or' a)) 'or' (('not' b
'or' c) '&' ('not' c 'or' b))) '&' ('not'( ('not' b 'or' c) '&' ('not' c 'or' b
)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:14
.=((('not'( 'not' a 'or' c) 'or' 'not'( 'not' c 'or' a)) 'or' (('not' b
'or' c) '&' ('not' c 'or' b))) '&' (('not'( 'not' b 'or' c) 'or' 'not'( 'not' c
'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:14
.=(((('not' 'not' a '&' 'not' c) 'or' 'not'( 'not' c 'or' a)) 'or' ((
'not' b 'or' c) '&' ('not' c 'or' b))) '&' (('not'( 'not' b 'or' c) 'or' 'not'(
'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by
BVFUNC_1:13
.=(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&' (('not'( 'not' b 'or' c) 'or'
'not'( 'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by
BVFUNC_1:13
.=(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or'
(('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' 'not' b '&' 'not' c) 'or'
'not'( 'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by
BVFUNC_1:13
.=((((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or' (('not' b 'or' c) '&' (
'not' c 'or' b))) '&' (((b '&' 'not' c) 'or' ('not' 'not' c '&' 'not' b)) 'or'
(('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:13
.=(((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or' (('not' b 'or' c) '&' (
'not' c 'or' b))).z '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or' (('not' a
'or' c) '&' ('not' c 'or' a))).z by MARGREL1:def 20
.=(((a '&' 'not' c) 'or' (c '&' 'not' a)).z 'or' (('not' b 'or' c) '&' (
'not' c 'or' b)).z) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or' (('not' a
'or' c) '&' ('not' c 'or' a))).z by BVFUNC_1:def 4
.=(((a '&' 'not' c) 'or' (c '&' 'not' a)).z 'or' (('not' b 'or' c) '&' (
'not' c 'or' b)).z) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)).z 'or' (('not'
a 'or' c) '&' ('not' c 'or' a)).z) by BVFUNC_1:def 4
.=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c) '&'
('not' c 'or' b)).z) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)).z 'or' (('not'
a 'or' c) '&' ('not' c 'or' a)).z) by BVFUNC_1:def 4
.=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z
'&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)).z 'or' ((
'not' a 'or' c) '&' ('not' c 'or' a)).z) by MARGREL1:def 20
.=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z
'&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' (
('not' a 'or' c) '&' ('not' c 'or' a)).z) by BVFUNC_1:def 4
.=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z
'&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' (
('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20
.=(((a.z '&' ('not' c).z) 'or' (c '&' 'not' a).z) 'or' (('not' b 'or'
c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z)
'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' (('not' b
'or' c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b)
.z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not'
b).z 'or' c.z) '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&'
'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by BVFUNC_1:def 4
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not'
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b '&' 'not' c).z 'or' (c
'&' 'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by
BVFUNC_1:def 4
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not'
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c '&' 'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by
MARGREL1:def 20
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not'
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' ('not' b).z)) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z))
by MARGREL1:def 20
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not'
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' ('not' c 'or'
a).z)) by BVFUNC_1:def 4
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not'
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z
'or' a.z))) by BVFUNC_1:def 4
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' ((('not'
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z
'or' a.z))) by MARGREL1:def 19
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' (('not' (
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z
'or' a.z))) by MARGREL1:def 19
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' (('not' (
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' 'not' b.z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z
'or' a.z))) by MARGREL1:def 19
.=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' (('not' (
b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' 'not' b.z)) 'or' (('not' a.z 'or' c.z) '&' (('not' c).z
'or' a.z))) by MARGREL1:def 19;
(b 'imp' a).z=TRUE by A2,A1,MARGREL1:12;
then
A8: 'not' b.z=TRUE or a.z=TRUE by A6,A4;
now
per cases by A3,A5;
case
A9: 'not' a.z=TRUE;
then a.z=FALSE;
then ((a 'eqv' c) 'eqv' (b 'eqv' c)).z =((FALSE 'or' (c.z '&' TRUE))
'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&'
('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or' c.z) '&' (('not'
c).z 'or' FALSE))) by A7
.=((FALSE 'or' c.z) 'or' (('not' b.z 'or' c.z) '&' (('not' c).
z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z))
'or' ((TRUE 'or' c.z) '&' (('not' c).z 'or' FALSE)))
.=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z)
)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or'
c.z) '&' (('not' c).z 'or' FALSE)))
.=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z)
)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or'
c.z) '&' ('not' c).z))
.=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z)
)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' (TRUE '&' (
'not' c).z))
.=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z)
)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z
)
.=((c.z 'or' (c.z 'or' 'not' b.z)) '&' (c.z 'or' (('not' c).
z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z))
'or' ('not' c).z) by XBOOLEAN:9
.=(((c.z 'or' c.z) 'or' 'not' b.z) '&' (c.z 'or' (('not' c).
z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z))
'or' ('not' c).z)
.=((c.z 'or' 'not' b.z) '&' ((c.z 'or' ('not' c).z) 'or' b.z
)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z
)
.=((c.z 'or' 'not' b.z) '&' ((c.z 'or' 'not' c.z) 'or' b.z
)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z
) by MARGREL1:def 19
.=((c.z 'or' 'not' b.z) '&' (TRUE 'or' b.z)) '&' (((b.z '&'
('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z) by XBOOLEAN:102
.=(TRUE '&' (c.z 'or' 'not' b.z)) '&' (((b.z '&' ('not' c).z)
'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z)
.=(c.z 'or' 'not' b.z) '&' (((b.z '&' ('not' c).z) 'or' (c.z
'&' 'not' b.z)) 'or' ('not' c).z)
.=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (('not'
c).z 'or' (c.z '&' 'not' b.z)))
.=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (((
'not' c).z 'or' c.z) '&' (('not' c).z 'or' 'not' b.z))) by XBOOLEAN:9
.=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (('not'
c.z 'or' c.z) '&' (('not' c).z 'or' 'not' b.z))) by MARGREL1:def 19
.=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (TRUE
'&' (('not' c).z 'or' 'not' b.z))) by XBOOLEAN:102
.=(c.z 'or' 'not' b.z) '&' ((('not' c).z 'or' 'not' b.z) 'or'
(b.z '&' ('not' c).z))
.=(c.z 'or' 'not' b.z) '&' (((('not' c).z 'or' 'not' b.z) 'or'
b.z) '&' ((('not' c).z 'or' 'not' b.z) 'or' ('not' c).z)) by XBOOLEAN:9
.=(c.z 'or' 'not' b.z) '&' ((('not' c).z 'or' ('not' b.z 'or'
b.z)) '&' ((('not' c).z 'or' 'not' b.z) 'or' ('not' c).z))
.=(c.z 'or' 'not' b.z) '&' ((('not' c).z 'or' TRUE) '&' ((('not'
c).z 'or' 'not' b.z) 'or' ('not' c).z)) by XBOOLEAN:102
.=(c.z 'or' 'not' b.z) '&' (TRUE '&' ((('not' c).z 'or' 'not' (b
).z) 'or' ('not' c).z))
.=(c.z 'or' 'not' b.z) '&' (('not' b.z 'or' ('not' c).z) 'or'
('not' c).z)
.=(c.z 'or' 'not' b.z) '&' ('not' b.z 'or' (('not' c).z 'or' (
'not' c).z))
.=('not' b.z '&' (c.z 'or' 'not' b.z)) 'or' ((c.z 'or' 'not'
b.z) '&' ('not' c).z) by XBOOLEAN:8
.=(('not' b.z '&' c.z) 'or' ('not' b.z '&' 'not' b.z)) 'or'
(('not' c).z '&' (c.z 'or' 'not' b.z)) by XBOOLEAN:8
.=(('not' b.z '&' c.z) 'or' 'not' b.z) 'or' (('not' c).z '&' (
c).z 'or' ('not' c).z '&' 'not' b.z) by XBOOLEAN:8
.=(('not' b.z '&' c.z) 'or' 'not' b.z) 'or' ((c.z '&' 'not'
c.z) 'or' (('not' c).z '&' 'not' b.z)) by MARGREL1:def 19
.=(('not' b.z '&' c.z) 'or' 'not' b.z) 'or' (FALSE 'or' ((
'not' c).z '&' 'not' b.z)) by XBOOLEAN:138
.=('not' b.z 'or' ('not' b.z '&' c.z)) 'or' (('not' c).z '&'
'not' b.z)
.='not' b.z 'or' (('not' b.z '&' c.z) 'or' (('not' c).z '&'
'not' b.z))
.=TRUE by A8,A9;
hence thesis;
end;
case
b.z=TRUE;
then 'not' b.z=FALSE;
then
((a 'eqv' c) 'eqv' (b 'eqv' c)).z =((('not' c).z 'or' (FALSE '&' (c
).z)) 'or' ((FALSE 'or' c.z) '&' (('not' c).z 'or' TRUE))) '&' ((('not' c).z
'or' (c.z '&' FALSE)) 'or' ((FALSE 'or' c.z) '&' (('not' c).z 'or' TRUE)))
by A2,A1,A6,A4,A7
.=((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' (('not' c).
z 'or' TRUE))) '&' ((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' ((
'not' c).z 'or' TRUE)))
.=((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' TRUE)) '&'
((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' TRUE))
.=((('not' c).z 'or' FALSE) 'or' (c.z '&' TRUE)) '&' ((('not' c).z
'or' FALSE) 'or' (c.z '&' TRUE))
.=(('not' c).z 'or' (TRUE '&' c.z)) '&' (('not' c).z 'or' (c.z
'&' TRUE))
.=(('not' c).z 'or' c.z) '&' (('not' c).z 'or' c.z)
.=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z) by
MARGREL1:def 19
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a
'imp' c) 'eqv' (b 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7
.=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20;
assume
A2: (a 'eqv' b).z=TRUE;
then (a 'imp' b).z=TRUE by A1,MARGREL1:12;
then
A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8;
(b 'imp' a).z=TRUE by A2,A1,MARGREL1:12;
then
A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8;
A5: ((a 'imp' c) 'eqv' (b 'imp' c)).z =(((a 'imp' c) 'imp' (b 'imp' c)) '&'
((b 'imp' c) 'imp' (a 'imp' c))).z by BVFUNC_4:7
.=((a 'imp' c) 'imp' (b 'imp' c)).z '&' ((b 'imp' c) 'imp' (a 'imp' c)).
z by MARGREL1:def 20
.=('not' (a 'imp' c).z 'or' (b 'imp' c).z) '&' ((b 'imp' c) 'imp' (a
'imp' c)).z by BVFUNC_1:def 8
.=('not' (a 'imp' c).z 'or' (b 'imp' c).z) '&' ('not' (b 'imp' c).z 'or'
(a 'imp' c).z) by BVFUNC_1:def 8
.=('not'( 'not' a.z 'or' c.z) 'or' (b 'imp' c).z) '&' ('not' (b
'imp' c).z 'or' (a 'imp' c).z) by BVFUNC_1:def 8
.=('not'( 'not' a.z 'or' c.z) 'or' ('not' b.z 'or' c.z)) '&' (
'not' (b 'imp' c).z 'or' (a 'imp' c).z) by BVFUNC_1:def 8
.=('not'( 'not' a.z 'or' c.z) 'or' ('not' b.z 'or' c.z)) '&' (
'not'( 'not' b.z 'or' c.z) 'or' (a 'imp' c).z) by BVFUNC_1:def 8
.=((a.z '&' 'not' c.z) 'or' ('not' b.z 'or' c.z)) '&' ((b.z
'&' 'not' c.z) 'or' ('not' a.z 'or' c.z)) by BVFUNC_1:def 8;
A6: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8;
A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
now
per cases by A3,A8;
case
A9: 'not' a.z=TRUE;
then
a.z=FALSE;
then 'not' b.z=TRUE by A4;
then ((a 'imp' c) 'eqv' (b 'imp' c)).z =(FALSE 'or' (TRUE 'or' c.z))
'&' (FALSE 'or' (TRUE 'or' c.z)) by A5,A9
.=(TRUE 'or' c.z) '&' (TRUE 'or' c.z)
.=TRUE;
hence thesis;
end;
case
b.z=TRUE;
then 'not' b.z=FALSE;
then
((a 'imp' c) 'eqv' (b 'imp' c)).z =('not' c.z 'or' (FALSE 'or' (c
).z)) '&' ('not' c.z 'or' (FALSE 'or' c.z)) by A2,A1,A7,A6,A5
.=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z)
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (c
'imp' a) 'eqv' (c 'imp' b)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7
.=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20;
assume
A2: (a 'eqv' b).z=TRUE;
then (a 'imp' b).z=TRUE by A1,MARGREL1:12;
then
A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8;
(b 'imp' a).z=TRUE by A2,A1,MARGREL1:12;
then
A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8;
A5: ((c 'imp' a) 'eqv' (c 'imp' b)).z =(((c 'imp' a) 'imp' (c 'imp' b)) '&'
((c 'imp' b) 'imp' (c 'imp' a))).z by BVFUNC_4:7
.=((c 'imp' a) 'imp' (c 'imp' b)).z '&' ((c 'imp' b) 'imp' (c 'imp' a)).
z by MARGREL1:def 20
.=('not' (c 'imp' a).z 'or' (c 'imp' b).z) '&' ((c 'imp' b) 'imp' (c
'imp' a)).z by BVFUNC_1:def 8
.=('not' (c 'imp' a).z 'or' (c 'imp' b).z) '&' ('not' (c 'imp' b).z 'or'
(c 'imp' a).z) by BVFUNC_1:def 8
.=('not'( 'not' c.z 'or' a.z) 'or' (c 'imp' b).z) '&' ('not' (c
'imp' b).z 'or' (c 'imp' a).z) by BVFUNC_1:def 8
.=('not'( 'not' c.z 'or' a.z) 'or' ('not' c.z 'or' b.z)) '&' (
'not' (c 'imp' b).z 'or' (c 'imp' a).z) by BVFUNC_1:def 8
.=('not'( 'not' c.z 'or' a.z) 'or' ('not' c.z 'or' b.z)) '&' (
'not'( 'not' c.z 'or' b.z) 'or' (c 'imp' a).z) by BVFUNC_1:def 8
.=((c.z '&' 'not' a.z) 'or' ('not' c.z 'or' b.z)) '&' ((c.z
'&' 'not' b.z) 'or' ('not' c.z 'or' a.z)) by BVFUNC_1:def 8;
A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8;
A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
now
per cases by A3,A8;
case
A9: 'not' a.z=TRUE;
then
a.z=FALSE;
then 'not' b.z=TRUE by A4;
then ((c 'imp' a) 'eqv' (c 'imp' b)).z =(c.z 'or' ('not' c.z 'or'
FALSE)) '&' (c.z 'or' ('not' c.z 'or' FALSE)) by A5,A9
.=(c.z 'or' 'not' c.z) '&' (c.z 'or' 'not' c.z)
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
case
b.z=TRUE;
then 'not' b.z=FALSE;
then ((c 'imp' a) 'eqv' (c 'imp' b)).z =(FALSE 'or' ('not' c.z 'or'
TRUE)) '&' (FALSE 'or' ('not' c.z 'or' TRUE)) by A2,A1,A7,A5,
MARGREL1:12
.=('not' c.z 'or' TRUE) '&' ('not' c.z 'or' TRUE)
.=TRUE;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a
'&' c) 'eqv' (b '&' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7
.=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20;
assume
A2: (a 'eqv' b).z=TRUE;
then (a 'imp' b).z=TRUE by A1,MARGREL1:12;
then
A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8;
(b 'imp' a).z=TRUE by A2,A1,MARGREL1:12;
then
A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8;
A5: ((a '&' c) 'eqv' (b '&' c)).z =(((a '&' c) 'imp' (b '&' c)) '&' ((b '&'
c) 'imp' (a '&' c))).z by BVFUNC_4:7
.=((a '&' c) 'imp' (b '&' c)).z '&' ((b '&' c) 'imp' (a '&' c)).z by
MARGREL1:def 20
.=('not' (a '&' c).z 'or' (b '&' c).z) '&' ((b '&' c) 'imp' (a '&' c)).z
by BVFUNC_1:def 8
.=('not' (a '&' c).z 'or' (b '&' c).z) '&' ('not' (b '&' c).z 'or' (a
'&' c).z) by BVFUNC_1:def 8
.=('not'( a.z '&' c.z) 'or' (b '&' c).z) '&' ('not' (b '&' c).z 'or'
(a '&' c).z) by MARGREL1:def 20
.=('not'( a.z '&' c.z) 'or' (b.z '&' c.z)) '&' ('not' (b '&' c).
z 'or' (a '&' c).z) by MARGREL1:def 20
.=('not'( a.z '&' c.z) 'or' (b.z '&' c.z)) '&' ('not'( b.z '&'
c.z) 'or' (a '&' c).z) by MARGREL1:def 20
.=(('not' a.z 'or' 'not' c.z) 'or' (b.z '&' c.z)) '&' (('not' (b
).z 'or' 'not' c.z) 'or' (a.z '&' c.z)) by MARGREL1:def 20;
A6: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8;
A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
now
per cases by A3,A8;
case
A9: 'not' a.z=TRUE;
then
a.z=FALSE;
then 'not' b.z=TRUE by A4;
then
((a '&' c) 'eqv' (b '&' c)).z =((TRUE 'or' 'not' c.z) 'or' FALSE)
'&' ((TRUE 'or' 'not' c.z) 'or' FALSE) by A5,A9
.=(TRUE 'or' 'not' c.z) '&' (TRUE 'or' 'not' c.z)
.=TRUE;
hence thesis;
end;
case
b.z=TRUE;
then 'not' b.z=FALSE;
then
((a '&' c) 'eqv' (b '&' c)).z =((FALSE 'or' 'not' c.z) 'or' c.z
) '&' ((FALSE 'or' 'not' c.z) 'or' c.z) by A2,A1,A7,A6,A5
.=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z)
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a
'or' c) 'eqv' (b 'or' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7
.=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20;
assume
A2: (a 'eqv' b).z=TRUE;
then (a 'imp' b).z=TRUE by A1,MARGREL1:12;
then
A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8;
(b 'imp' a).z=TRUE by A2,A1,MARGREL1:12;
then
A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8;
A5: ((a 'or' c) 'eqv' (b 'or' c)).z =(((a 'or' c) 'imp' (b 'or' c)) '&' ((b
'or' c) 'imp' (a 'or' c))).z by BVFUNC_4:7
.=((a 'or' c) 'imp' (b 'or' c)).z '&' ((b 'or' c) 'imp' (a 'or' c)).z by
MARGREL1:def 20
.=('not' (a 'or' c).z 'or' (b 'or' c).z) '&' ((b 'or' c) 'imp' (a 'or' c
)).z by BVFUNC_1:def 8
.=('not' (a 'or' c).z 'or' (b 'or' c).z) '&' ('not' (b 'or' c).z 'or' (a
'or' c).z) by BVFUNC_1:def 8
.=('not'( a.z 'or' c.z) 'or' (b 'or' c).z) '&' ('not' (b 'or' c).z
'or' (a 'or' c).z) by BVFUNC_1:def 4
.=('not'( a.z 'or' c.z) 'or' (b.z 'or' c.z)) '&' ('not' (b 'or'
c).z 'or' (a 'or' c).z) by BVFUNC_1:def 4
.=('not'( a.z 'or' c.z) 'or' (b.z 'or' c.z)) '&' ('not'( b.z
'or' c.z) 'or' (a 'or' c).z) by BVFUNC_1:def 4
.=(('not' a.z '&' 'not' c.z) 'or' (b.z 'or' c.z)) '&' (('not' (b
).z '&' 'not' c.z) 'or' (a.z 'or' c.z)) by BVFUNC_1:def 4;
A6: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8;
A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
now
per cases by A3,A8;
case
A9: 'not' a.z=TRUE;
then
a.z=FALSE;
then 'not' b.z=TRUE by A4;
then
((a 'or' c) 'eqv' (b 'or' c)).z =('not' c.z 'or' (FALSE 'or' c.
z)) '&' ('not' c.z 'or' (FALSE 'or' c.z)) by A5,A9
.=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z)
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
case
b.z=TRUE;
then 'not' b.z=FALSE;
then
((a 'or' c) 'eqv' (b 'or' c)).z =(FALSE 'or' (TRUE 'or' c.z)) '&'
(FALSE 'or' (TRUE 'or' c.z)) by A2,A1,A7,A6,A5
.=(TRUE 'or' c.z) '&' (TRUE 'or' c.z)
.=TRUE;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '<' (a 'eqv' b)
'eqv' (b 'eqv' a) 'eqv' a
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ('not' a).z='not' a.z by MARGREL1:def 19;
assume
A2: a.z=TRUE;
then
A3: 'not' a.z=FALSE;
A4: ((a 'eqv' b) 'eqv' (b 'eqv' a)).z =(((a 'eqv' b) 'imp' (b 'eqv' a)) '&'
((b 'eqv' a) 'imp' (a 'eqv' b))).z by BVFUNC_4:7
.=('not'( a 'eqv' b) 'or' (b 'eqv' a)).z '&' ('not'( b 'eqv' a) 'or' (a
'eqv' b)).z by BVFUNC_4:8
.=(('not'( a 'eqv' b)).z 'or' (b 'eqv' a).z) '&' (('not'( b 'eqv' a)).z
'or' (a 'eqv' b).z) by BVFUNC_1:def 4
.=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' (b 'eqv' a).z) '&' ((
'not'( b 'eqv' a)).z 'or' (a 'eqv' b).z) by BVFUNC_4:7
.=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' (b 'eqv' a).z) '&' ((
'not'( b 'eqv' a)).z 'or' ((a 'imp' b) '&' (b 'imp' a)).z) by BVFUNC_4:7
.=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' ((b 'imp' a) '&' (a
'imp' b)).z) '&' (('not'( b 'eqv' a)).z 'or' ((a 'imp' b) '&' (b 'imp' a)).z)
by BVFUNC_4:7
.=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' ((b 'imp' a) '&' (a
'imp' b)).z) '&' (('not'( (b 'imp' a) '&' (a 'imp' b))).z 'or' ((a 'imp' b) '&'
(b 'imp' a)).z) by BVFUNC_4:7
.=('not' ((a 'imp' b) '&' (b 'imp' a)).z 'or' ((b 'imp' a) '&' (a 'imp'
b)).z) '&' ('not' ((b 'imp' a) '&' (a 'imp' b)).z 'or' ((a 'imp' b) '&' (b
'imp' a)).z) by MARGREL1:def 19
.=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a) '&' (a
'imp' b)).z) '&' ('not' ((b 'imp' a) '&' (a 'imp' b)).z 'or' ((a 'imp' b) '&' (
b 'imp' a)).z) by MARGREL1:def 20
.=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a
'imp' b).z)) '&' ('not' ((b 'imp' a) '&' (a 'imp' b)).z 'or' ((a 'imp' b) '&' (
b 'imp' a)).z) by MARGREL1:def 20
.=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a
'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b) '&'
(b 'imp' a)).z) by MARGREL1:def 20
.=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a
'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).z
'&' (b 'imp' a).z)) by MARGREL1:def 20
.=('not'( ('not' a 'or' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&'
(a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).
z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' ((b 'imp' a).z
'&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp'
b).z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or'
a).z '&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a
'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or'
a).z '&' ('not' a 'or' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or'
((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or'
a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' (a 'imp' b).z)
'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or'
a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' ('not' a 'or'
b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or'
a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' ('not' a 'or'
b).z) 'or' (('not' a 'or' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8
.=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or'
a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' ('not' a 'or'
b).z) 'or' (('not' a 'or' b).z '&' ('not' b 'or' a).z)) by BVFUNC_4:8
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b 'or' a).z)) 'or' (((
'not' b 'or' a).z) '&' (('not' a 'or' b).z))) '&' ('not'( (('not' b 'or' a).z)
'&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a).z)))
by BVFUNC_1:def 4
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' ((
('not' b 'or' a).z) '&' (('not' a 'or' b).z))) '&' ('not'( (('not' b 'or' a).z)
'&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a).z)))
by BVFUNC_1:def 4
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' ((
('not' b).z 'or' a.z) '&' (('not' a 'or' b).z))) '&' ('not'( (('not' b 'or' a
).z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a)
.z))) by BVFUNC_1:def 4
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' ((
('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b
'or' a).z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b
'or' a).z))) by BVFUNC_1:def 4
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' ((
('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b).z
'or' a.z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b
'or' a).z))) by BVFUNC_1:def 4
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' ((
('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b).z
'or' a.z) '&' (('not' a).z 'or' b.z)) 'or' ((('not' a 'or' b).z) '&' ((
'not' b 'or' a).z))) by BVFUNC_1:def 4
.=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' ((
('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b).z
'or' a.z) '&' (('not' a).z 'or' b.z)) 'or' ((('not' a).z 'or' b.z) '&' ((
'not' b 'or' a).z))) by BVFUNC_1:def 4
.=('not'( (FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE)) 'or' ((('not'
b).z 'or' TRUE) '&' (FALSE 'or' b.z))) '&' ('not'( (('not' b).z 'or' TRUE)
'&' (FALSE 'or' b.z)) 'or' ((FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE)))
by A3,A1,BVFUNC_1:def 4
.=('not'( b.z '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or'
TRUE) '&' (FALSE 'or' b.z))) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE
'or' b.z)) 'or' ((FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE)))
.=('not'( b.z '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or'
TRUE) '&' b.z)) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z))
'or' ((FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE)))
.=('not'( b.z '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or'
TRUE) '&' b.z)) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z))
'or' (b.z '&' (('not' b).z 'or' TRUE)))
.=('not'( b.z '&' TRUE) 'or' ((('not' b).z 'or' TRUE) '&' b.z)) '&'
('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' (b.z '&' (('not'
b).z 'or' TRUE)))
.=('not'( b.z '&' TRUE) 'or' (TRUE '&' b.z)) '&' ('not'( (('not' b).
z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' (b.z '&' (('not' b).z 'or' TRUE)))
.=('not'( b.z '&' TRUE) 'or' (TRUE '&' b.z)) '&' ('not'( TRUE '&' (
FALSE 'or' b.z)) 'or' (b.z '&' (('not' b).z 'or' TRUE)))
.=('not'( b.z '&' TRUE) 'or' (TRUE '&' b.z)) '&' ('not'( TRUE '&' (
FALSE 'or' b.z)) 'or' (b.z '&' TRUE))
.=('not'( TRUE '&' b.z) 'or' (TRUE '&' b.z)) '&' ('not'( TRUE '&' (b
).z) 'or' (b.z '&' TRUE))
.=('not' b.z 'or' (TRUE '&' b.z)) '&' ('not' b.z 'or' (TRUE '&' (b
).z))
.=('not' b.z 'or' b.z) '&' ('not' b.z 'or' b.z)
.=TRUE by XBOOLEAN:102;
((a 'eqv' b) 'eqv' (b 'eqv' a) 'eqv' a).z =((((a 'eqv' b) 'eqv' (b 'eqv'
a)) 'imp' a) '&' (a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a)))).z by BVFUNC_4:7
.=(((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a).z '&' (a 'imp' ((a 'eqv' b)
'eqv' (b 'eqv' a))).z by MARGREL1:def 20
.=('not' ((a 'eqv' b) 'eqv' (b 'eqv' a)).z 'or' a.z) '&' (a 'imp' ((a
'eqv' b) 'eqv' (b 'eqv' a))).z by BVFUNC_1:def 8
.=('not' ((a 'eqv' b) 'eqv' (b 'eqv' a)).z 'or' a.z) '&' ('not' a.z
'or' ((a 'eqv' b) 'eqv' (b 'eqv' a)).z) by BVFUNC_1:def 8
.=(FALSE 'or' a.z) '&' ('not' a.z 'or' TRUE) by A4
.=a.z '&' ('not' a.z 'or' TRUE)
.=TRUE '&' a.z
.=TRUE by A2;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '<' (a 'imp' b) 'eqv' b
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
assume
a.z=TRUE;
then
A2: 'not' a.z=FALSE;
((a 'imp' b) 'eqv' b).z =(('not' a 'or' b) 'eqv' b).z by BVFUNC_4:8
.=((('not' a 'or' b) 'imp' b) '&' (b 'imp' ('not' a 'or' b))).z by
BVFUNC_4:7
.=(('not'( 'not' a 'or' b) 'or' b) '&' (b 'imp' ('not' a 'or' b))).z by
BVFUNC_4:8
.=(('not'( 'not' a 'or' b) 'or' b) '&' ('not' b 'or' ('not' a 'or' b))).
z by BVFUNC_4:8
.=('not'( 'not' a 'or' b) 'or' b).z '&' ('not' b 'or' ('not' a 'or' b)).
z by MARGREL1:def 20
.=(('not'( 'not' a 'or' b)).z 'or' b.z) '&' ('not' b 'or' ('not' a
'or' b)).z by BVFUNC_1:def 4
.=('not' ('not' a 'or' b).z 'or' b.z) '&' ('not' b 'or' ('not' a 'or'
b)).z by MARGREL1:def 19
.=('not'( ('not' a).z 'or' b.z) 'or' b.z) '&' ('not' b 'or' ('not' a
'or' b)).z by BVFUNC_1:def 4
.=(('not' 'not' a.z '&' 'not' b.z) 'or' b.z) '&' ('not' b 'or' (
'not' a 'or' b)).z by MARGREL1:def 19
.=((a.z '&' 'not' b.z) 'or' b.z) '&' (('not' b).z 'or' ('not' a
'or' b).z) by BVFUNC_1:def 4
.=((a.z '&' 'not' b.z) 'or' b.z) '&' (('not' b).z 'or' (('not' a).
z 'or' b.z)) by BVFUNC_1:def 4
.=((a.z '&' 'not' b.z) 'or' b.z) '&' (('not' b).z 'or' ('not' a.
z 'or' b.z)) by MARGREL1:def 19
.=((TRUE '&' 'not' b.z) 'or' b.z) '&' ('not' b.z 'or' (FALSE 'or'
b.z)) by A2,MARGREL1:def 19
.=('not' b.z 'or' b.z) '&' ('not' b.z 'or' (FALSE 'or' b.z))
.=('not' b.z 'or' b.z) '&' ('not' b.z 'or' b.z)
.=TRUE by XBOOLEAN:102;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '<' (b 'imp' a) 'eqv' a
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
assume
a.z=TRUE;
then
A2: 'not' a.z=FALSE;
((b 'imp' a) 'eqv' a).z =(('not' b 'or' a) 'eqv' a).z by BVFUNC_4:8
.=((('not' b 'or' a) 'imp' a) '&' (a 'imp' ('not' b 'or' a))).z by
BVFUNC_4:7
.=(('not'( 'not' b 'or' a) 'or' a) '&' (a 'imp' ('not' b 'or' a))).z by
BVFUNC_4:8
.=(('not'( 'not' b 'or' a) 'or' a) '&' ('not' a 'or' ('not' b 'or' a))).
z by BVFUNC_4:8
.=('not'( 'not' b 'or' a) 'or' a).z '&' ('not' a 'or' ('not' b 'or' a)).
z by MARGREL1:def 20
.=(('not'( 'not' b 'or' a)).z 'or' a.z) '&' ('not' a 'or' ('not' b
'or' a)).z by BVFUNC_1:def 4
.=('not' ('not' b 'or' a).z 'or' a.z) '&' ('not' a 'or' ('not' b 'or'
a)).z by MARGREL1:def 19
.=('not'( ('not' b).z 'or' a.z) 'or' a.z) '&' ('not' a 'or' ('not' b
'or' a)).z by BVFUNC_1:def 4
.=(('not' 'not' b.z '&' 'not' a.z) 'or' a.z) '&' ('not' a 'or' (
'not' b 'or' a)).z by MARGREL1:def 19
.=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' ('not' b
'or' a).z) by BVFUNC_1:def 4
.=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' (('not' b).
z 'or' a.z)) by BVFUNC_1:def 4
.=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' ('not' b.
z 'or' a.z)) by MARGREL1:def 19
.=((b.z '&' 'not' a.z) 'or' a.z) '&' ('not' a.z 'or' ('not' b.
z 'or' a.z)) by MARGREL1:def 19
.=TRUE '&' (FALSE 'or' ('not' b.z 'or' TRUE)) by A2
.=FALSE 'or' ('not' b.z 'or' TRUE)
.='not' b.z 'or' TRUE
.=TRUE;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '<' (a '&' b) 'eqv'
(b '&' a) 'eqv' a
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
assume
A1: a.z=TRUE;
A2: ((a '&' b) 'eqv' (a '&' b)).z =(((a '&' b) 'imp' (a '&' b)) '&' ((a '&'
b) 'imp' (a '&' b))).z by BVFUNC_4:7
.=('not'( a '&' b) 'or' (a '&' b)).z by BVFUNC_4:8
.=(I_el(Y)).z by BVFUNC_4:6
.=TRUE by BVFUNC_1:def 11;
((a '&' b) 'eqv' (b '&' a) 'eqv' a).z =((((a '&' b) 'eqv' (a '&' b))
'imp' a) '&' (a 'imp' ((a '&' b) 'eqv' (a '&' b)))).z by BVFUNC_4:7
.=(((a '&' b) 'eqv' (a '&' b)) 'imp' a).z '&' (a 'imp' ((a '&' b) 'eqv'
(a '&' b))).z by MARGREL1:def 20
.=('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a).z '&' (a 'imp' ((a '&' b)
'eqv' (a '&' b))).z by BVFUNC_4:8
.=('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a).z '&' ('not' a 'or' ((a '&'
b) 'eqv' (a '&' b))).z by BVFUNC_4:8
.=(('not'( (a '&' b) 'eqv' (a '&' b))).z 'or' a.z) '&' ('not' a 'or' (
(a '&' b) 'eqv' (a '&' b))).z by BVFUNC_1:def 4
.=(('not'( (a '&' b) 'eqv' (a '&' b))).z 'or' a.z) '&' (('not' a).z
'or' ((a '&' b) 'eqv' (a '&' b)).z) by BVFUNC_1:def 4
.=('not' ((a '&' b) 'eqv' (a '&' b)).z 'or' a.z) '&' (('not' a).z 'or'
((a '&' b) 'eqv' (a '&' b)).z) by MARGREL1:def 19
.=(FALSE 'or' a.z) '&' (('not' a).z 'or' TRUE) by A2
.=a.z '&' (('not' a).z 'or' TRUE)
.=TRUE '&' a.z
.=TRUE by A1;
hence thesis;
end;
begin :: BVFUNC_8
reserve Y for non empty set;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds a 'imp' (b '&' c
'&' d) = (a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d)).x =((a 'imp' b) '&' (a
'imp' c)).x '&' (a 'imp' d).x by MARGREL1:def 20
.=(a 'imp' b).x '&' (a 'imp' c).x '&' (a 'imp' d).x by MARGREL1:def 20
.=('not' a.x 'or' b.x) '&' (a 'imp' c).x '&' (a 'imp' d).x by
BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' a.x 'or' c.x) '&' (a 'imp' d
).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) '&' ('not' a.x 'or' c.x) '&' ('not' (a)
.x 'or' (d).x) by BVFUNC_1:def 8
.=('not' a.x 'or' (b.x '&' c.x)) '&' ('not' a.x 'or' (d).x) by
XBOOLEAN:9
.='not' a.x 'or' (b.x '&' c.x '&' (d).x) by XBOOLEAN:9
.='not' a.x 'or' ((b '&' c).x '&' (d).x) by MARGREL1:def 20
.='not' a.x 'or' ((b '&' c '&' d).x) by MARGREL1:def 20
.=(a 'imp' (b '&' c '&' d)).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds a 'imp' (b 'or' c
'or' d) = (a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d)).x =((a 'imp' b) 'or' (
a 'imp' c)).x 'or' (a 'imp' d).x by BVFUNC_1:def 4
.=(a 'imp' b).x 'or' (a 'imp' c).x 'or' (a 'imp' d).x by BVFUNC_1:def 4
.=('not' a.x 'or' b.x) 'or' (a 'imp' c).x 'or' (a 'imp' d).x by
BVFUNC_1:def 8
.=('not' a.x 'or' b.x) 'or' ('not' a.x 'or' c.x) 'or' (a 'imp'
d).x by BVFUNC_1:def 8
.=('not' a.x 'or' b.x) 'or' ('not' a.x 'or' c.x) 'or' ('not' (
a).x 'or' (d).x) by BVFUNC_1:def 8
.=(('not' a.x 'or' ('not' a.x 'or' b.x)) 'or' c.x) 'or' ('not'
a.x 'or' (d).x)
.=((('not' a.x 'or' 'not' a.x) 'or' b.x) 'or' c.x) 'or' ('not'
a.x 'or' (d).x) by BINARITH:11
.=('not' a.x 'or' (b.x 'or' c.x)) 'or' ('not' a.x 'or' (d).x)
.=('not' a.x 'or' (b 'or' c).x) 'or' ('not' a.x 'or' (d).x) by
BVFUNC_1:def 4
.=('not' a.x 'or' ('not' a.x 'or' (b 'or' c).x)) 'or' (d).x
.=(('not' a.x 'or' 'not' a.x) 'or' (b 'or' c).x) 'or' (d).x by
BINARITH:11
.='not' a.x 'or' ((b 'or' c).x 'or' (d).x)
.='not' a.x 'or' (b 'or' c 'or' d).x by BVFUNC_1:def 4
.=(a 'imp' (b 'or' c 'or' d)).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds (a '&' b '&' c)
'imp' d = (a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d)).x =((a 'imp' d) 'or' (
b 'imp' d)).x 'or' (c 'imp' d).x by BVFUNC_1:def 4
.=(a 'imp' d).x 'or' (b 'imp' d).x 'or' (c 'imp' d).x by BVFUNC_1:def 4
.=('not' a.x 'or' (d).x) 'or' (b 'imp' d).x 'or' (c 'imp' d).x by
BVFUNC_1:def 8
.=('not' a.x 'or' (d).x) 'or' ('not' b.x 'or' (d).x) 'or' (c 'imp'
d).x by BVFUNC_1:def 8
.=('not' a.x 'or' (d).x) 'or' ('not' b.x 'or' (d).x) 'or' ('not' (
c).x 'or' (d).x) by BVFUNC_1:def 8
.=(('not' a.x 'or' ((d).x 'or' 'not' b.x)) 'or' (d).x) 'or' ('not'
c.x 'or' (d).x)
.=(('not' a.x 'or' 'not' b.x) 'or' (d).x) 'or' (d).x 'or' ('not' (
c).x 'or' (d).x)
.=('not' a.x 'or' 'not' b.x) 'or' ((d).x 'or' (d).x) 'or' ('not' (
c).x 'or' (d).x) by BINARITH:11
.=('not'( a.x '&' b.x) 'or' ((d).x 'or' 'not' c.x)) 'or' (d).x
.=(('not'( a.x '&' b.x) 'or' 'not' c.x) 'or' (d).x) 'or' (d).x
.=('not'( a.x '&' b.x) 'or' 'not' c.x) 'or' ((d).x 'or' (d).x)
by BINARITH:11
.='not'( (a '&' b).x '&' c.x) 'or' (d).x by MARGREL1:def 20
.='not' (a '&' b '&' c).x 'or' (d).x by MARGREL1:def 20
.=((a '&' b '&' c) 'imp' d).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds (a 'or' b 'or' c)
'imp' d = (a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d)).x =((a 'imp' d) '&' (b
'imp' d)).x '&' (c 'imp' d).x by MARGREL1:def 20
.=(a 'imp' d).x '&' (b 'imp' d).x '&' (c 'imp' d).x by MARGREL1:def 20
.=('not' a.x 'or' (d).x) '&' (b 'imp' d).x '&' (c 'imp' d).x by
BVFUNC_1:def 8
.=('not' a.x 'or' (d).x) '&' ('not' b.x 'or' (d).x) '&' (c 'imp' d
).x by BVFUNC_1:def 8
.=((d).x 'or' 'not' a.x) '&' ('not' b.x 'or' (d).x) '&' ('not' (c)
.x 'or' (d).x) by BVFUNC_1:def 8
.=('not'( a.x 'or' b.x) 'or' (d).x) '&' ('not' c.x 'or' (d).x)
by XBOOLEAN:9
.=((d).x 'or' 'not' (a 'or' b).x) '&' ('not' c.x 'or' (d).x) by
BVFUNC_1:def 4
.=('not'( (a 'or' b).x 'or' c.x)) 'or' (d).x by XBOOLEAN:9
.='not' (a 'or' b 'or' c).x 'or' (d).x by BVFUNC_1:def 4
.=((a 'or' b 'or' c) 'imp' d).x by BVFUNC_1:def 8;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '&' (c 'imp' a) = (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b
'imp' a) '&' (a 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) =('not' a 'or' b) '&' (b
'imp' c) '&' (c 'imp' a) by BVFUNC_4:8
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (c 'imp' a) by BVFUNC_4:8
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a) by
BVFUNC_4:8
.=(('not' a '&' ('not' b 'or' c) 'or' b '&' ('not' b 'or' c))) '&' (
'not' c 'or' a) by BVFUNC_1:12
.=(('not' a '&' ('not' b 'or' c) 'or' (b '&' 'not' b 'or' b '&' c)))
'&' ('not' c 'or' a) by BVFUNC_1:12
.=(('not' a '&' ('not' b 'or' c) 'or' (O_el(Y) 'or' b '&' c))) '&' (
'not' c 'or' a) by BVFUNC_4:5
.=(('not' a '&' ('not' b 'or' c) 'or' (b '&' c))) '&' ('not' c 'or' a)
by BVFUNC_1:9
.=(('not' a 'or' (b '&' c)) '&' (('not' b 'or' c) 'or' (b '&' c))) '&'
('not' c 'or' a) by BVFUNC_1:11
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c) 'or'
(b '&' c))) '&' ('not' c 'or' a) by BVFUNC_1:11
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((('not' b 'or' c) 'or'
b) '&' (('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:11
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((c 'or' ('not' b 'or'
b)) '&' (('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:8
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((c 'or' I_el(Y)) '&' (
('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_4:6
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (I_el(Y) '&' (('not' b
'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:10
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c) 'or'
c)) '&' ('not' c 'or' a) by BVFUNC_1:6
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ('not' b 'or' (c 'or' c
))) '&' ('not' c 'or' a) by BVFUNC_1:8
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c) '&' (
'not' c 'or' a)) by BVFUNC_1:4
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c
'or' a)) 'or' (c '&' ('not' c 'or' a))) by BVFUNC_1:12
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c
'or' a)) 'or' ((c '&' 'not' c 'or' c '&' a))) by BVFUNC_1:12
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c
'or' a)) 'or' ((O_el(Y) 'or' c '&' a))) by BVFUNC_4:5
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c
'or' a)) 'or' (c '&' a)) by BVFUNC_1:9
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' (c '&' a)
) '&' (('not' c 'or' a) 'or' (c '&' a))) by BVFUNC_1:11
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((('not' b 'or' c) '&' (
'not' b 'or' a)) '&' (('not' c 'or' a) 'or' (c '&' a))) by BVFUNC_1:11
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&'
('not' b 'or' c)) '&' ((('not' c 'or' a) 'or' c) '&' (('not' c 'or' a) 'or' a))
) by BVFUNC_1:11
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&'
('not' b 'or' c)) '&' ((a 'or' ('not' c 'or' c)) '&' (('not' c 'or' a) 'or' a))
) by BVFUNC_1:8
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&'
('not' b 'or' c)) '&' ((a 'or' I_el(Y)) '&' (('not' c 'or' a) 'or' a))) by
BVFUNC_4:6
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&'
('not' b 'or' c)) '&' (I_el(Y) '&' (('not' c 'or' a) 'or' a))) by BVFUNC_1:10
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&'
('not' b 'or' c)) '&' (('not' c 'or' a) 'or' a)) by BVFUNC_1:6
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&'
('not' b 'or' c)) '&' ('not' c 'or' (a 'or' a))) by BVFUNC_1:8
.= ((('not' a 'or' c) '&' ('not' a 'or' b)) '&' (('not' b 'or' a) '&'
('not' b 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:4
.= (((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ('not' b 'or' a)) '&'
('not' b 'or' c)) '&' ('not' c 'or' a) by BVFUNC_1:4
.= (((('not' b 'or' a) '&' ('not' a 'or' c)) '&' ('not' a 'or' b)) '&'
('not' b 'or' c)) '&' ('not' c 'or' a) by BVFUNC_1:4
.= ((('not' b 'or' a) '&' ('not' a 'or' c)) '&' (('not' a 'or' b) '&'
('not' b 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:4
.= (('not' b 'or' a) '&' ('not' a 'or' c)) '&' ((('not' a 'or' b) '&'
('not' b 'or' c)) '&' ('not' c 'or' a)) by BVFUNC_1:4
.= ('not' b 'or' a) '&' (('not' a 'or' b) '&' ('not' b 'or' c) '&' (
'not' c 'or' a)) '&' ('not' a 'or' c) by BVFUNC_1:4
.= (a 'imp' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a) '&' ('not' b
'or' a) '&' ('not' a 'or' c) by BVFUNC_4:8
.= (a 'imp' b) '&' (b 'imp' c) '&' ('not' c 'or' a) '&' ('not' b 'or'
a) '&' ('not' a 'or' c) by BVFUNC_4:8
.= (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' ('not' b 'or' a)
'&' ('not' a 'or' c) by BVFUNC_4:8
.= (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b 'imp' a) '&' (
'not' a 'or' c) by BVFUNC_4:8
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b 'imp' a) '&' (a
'imp' c) by BVFUNC_4:8;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a = (a '&' b) 'or' (a
'&' 'not' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
((a '&' b) 'or' (a '&' 'not' b)).x =(a '&' (b 'or' 'not' b)).x by
BVFUNC_1:12
.=(a '&' I_el(Y)).x by BVFUNC_4:6
.=a.x by BVFUNC_1:6;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a = (a 'or' b) '&' (a
'or' 'not' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'or' b) '&' (a 'or' 'not' b)).x =(a 'or' (b '&' 'not' b)).x by
BVFUNC_1:11
.=(a 'or' O_el(Y)).x by BVFUNC_4:5
.=a.x by BVFUNC_1:9;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds a = (a '&' b '&' c)
'or' (a '&' b '&' 'not' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&'
'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a '&' b '&' c) 'or' (a '&' b '&' 'not' c) 'or' (a '&' 'not' b '&' c)
'or' (a '&' 'not' b '&' 'not' c)).x =(((a '&' b) '&' (c 'or' 'not' c)) 'or' (a
'&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c)).x by BVFUNC_1:12
.=(((a '&' b) '&' I_el(Y)) 'or' (a '&' 'not' b '&' c) 'or' (a '&'
'not' b '&' 'not' c)).x by BVFUNC_4:6
.=((a '&' b) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not'
c)).x by BVFUNC_1:6
.=((a '&' b) 'or' ((a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not'
c))).x by BVFUNC_1:8
.=((a '&' b) 'or' ((a '&' 'not' b) '&' (c 'or' 'not' c))).x by
BVFUNC_1:12
.=((a '&' b) 'or' ((a '&' 'not' b) '&' I_el(Y))).x by BVFUNC_4:6
.=((a '&' b) 'or' (a '&' 'not' b)).x by BVFUNC_1:6
.=(a '&' (b 'or' 'not' b)).x by BVFUNC_1:12
.=(a '&' I_el(Y)).x by BVFUNC_4:6
.=a.x by BVFUNC_1:6;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds a = (a 'or' b 'or' c
) '&' (a 'or' b 'or' 'not' c) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b
'or' 'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'or' b 'or' c) '&' (a 'or' b 'or' 'not' c) '&' (a 'or' 'not' b
'or' c) '&' (a 'or' 'not' b 'or' 'not' c)).x =(((a 'or' b) 'or' (c '&' 'not' c)
) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c)).x by
BVFUNC_1:11
.=(((a 'or' b) 'or' O_el(Y)) '&' (a 'or' 'not' b 'or' c) '&' (a 'or'
'not' b 'or' 'not' c)).x by BVFUNC_4:5
.=((a 'or' b) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or'
'not' c)).x by BVFUNC_1:9
.=((a 'or' b) '&' ((a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or'
'not' c))).x by BVFUNC_1:4
.=((a 'or' b) '&' ((a 'or' 'not' b) 'or' (c '&' 'not' c))).x by
BVFUNC_1:11
.=((a 'or' b) '&' ((a 'or' 'not' b) 'or' O_el(Y))).x by BVFUNC_4:5
.=((a 'or' b) '&' (a 'or' 'not' b)).x by BVFUNC_1:9
.=(a 'or' (b '&' 'not' b)).x by BVFUNC_1:11
.=(a 'or' O_el(Y)).x by BVFUNC_4:5
.=a.x by BVFUNC_1:9;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '&' b = a '&' ('not'
a 'or' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
(a '&' ('not' a 'or' b)).x =a.x '&' ('not' a 'or' b).x by MARGREL1:def 20
.=a.x '&' (('not' a).x 'or' b.x) by BVFUNC_1:def 4
.=a.x '&' ('not' a).x 'or' a.x '&' b.x by XBOOLEAN:8
.=a.x '&' 'not' a.x 'or' a.x '&' b.x by MARGREL1:def 19
.=FALSE 'or' a.x '&' b.x by XBOOLEAN:138
.=a.x '&' b.x
.=(a '&' b).x by MARGREL1:def 20;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a 'or' b = a 'or' (
'not' a '&' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'or' ('not' a '&' b)).x =a.x 'or' ('not' a '&' b).x by BVFUNC_1:def 4
.=a.x 'or' (('not' a).x '&' b.x) by MARGREL1:def 20
.=(a.x 'or' ('not' a).x) '&' (a.x 'or' b.x) by XBOOLEAN:9
.=(a.x 'or' 'not' a.x) '&' (a.x 'or' b.x) by MARGREL1:def 19
.=TRUE '&' (a.x 'or' b.x) by XBOOLEAN:102
.=a.x 'or' b.x
.=(a 'or' b).x by BVFUNC_1:def 4;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a 'xor' b = 'not'( a 'eqv' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'xor' b).x =('not' 'not'( ('not' a '&' b) 'or' (a '&' 'not' b))).x
by BVFUNC_4:9
.=('not'('not'('not' a '&' b) '&' 'not'( a '&' 'not' b))).x by
BVFUNC_1:13
.=('not'( ('not' 'not' a 'or' 'not' b) '&' 'not'( a '&' 'not' b))).x
by BVFUNC_1:14
.=('not'( (a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b))).x by
BVFUNC_1:14
.=('not'( (b 'imp' a) '&' ('not' a 'or' b))).x by BVFUNC_4:8
.=('not'( (b 'imp' a) '&' (a 'imp' b))).x by BVFUNC_4:8
.=('not'( a 'eqv' b)).x by BVFUNC_4:7;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a 'xor' b = (a 'or' b)
'&' ('not' a 'or' 'not' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'or' b) '&' ('not' a 'or' 'not' b)).x =(a 'or' b).x '&' ('not' a
'or' 'not' b).x by MARGREL1:def 20
.=(a.x 'or' b.x) '&' ('not' a 'or' 'not' b).x by BVFUNC_1:def 4
.=(a.x 'or' b.x) '&' (('not' a).x 'or' ('not' b).x) by BVFUNC_1:def 4
.=(('not' a).x '&' (a.x 'or' b.x)) 'or' ((a.x 'or' b.x) '&' (
'not' b).x) by XBOOLEAN:8
.=(('not' a).x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x
'&' (a.x 'or' b.x)) by XBOOLEAN:8
.=(('not' a).x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x
'&' a.x 'or' ('not' b).x '&' b.x) by XBOOLEAN:8
.=('not' a.x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x
'&' a.x 'or' ('not' b).x '&' b.x) by MARGREL1:def 19
.=('not' a.x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x
'&' a.x 'or' 'not' b.x '&' b.x) by MARGREL1:def 19
.=(FALSE 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or'
'not' b.x '&' b.x) by XBOOLEAN:138
.=(FALSE 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or'
FALSE) by XBOOLEAN:138
.=(('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' FALSE)
.=(('not' a).x '&' b.x) 'or' (a.x '&' ('not' b).x)
.=('not' a '&' b).x 'or' (a.x '&' ('not' b).x) by MARGREL1:def 20
.=('not' a '&' b).x 'or' (a '&' 'not' b).x by MARGREL1:def 20
.=('not' a '&' b 'or' a '&' 'not' b).x by BVFUNC_1:def 4
.=(a 'xor' b).x by BVFUNC_4:9;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'xor' I_el(Y) = 'not' a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'xor' I_el(Y)).x =(('not' a '&' I_el(Y)) 'or' (a '&' 'not' I_el(Y))
).x by BVFUNC_4:9
.=(('not' a '&' I_el(Y)) 'or' (a '&' O_el(Y))).x by BVFUNC_1:2
.=(('not' a '&' I_el(Y)) 'or' O_el(Y)).x by BVFUNC_1:5
.=('not' a '&' I_el(Y)).x by BVFUNC_1:9
.=('not' a).x by BVFUNC_1:6;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'xor' O_el(Y) = a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'xor' O_el(Y)).x =(('not' a '&' O_el(Y)) 'or' (a '&' 'not' O_el(Y))
).x by BVFUNC_4:9
.=(('not' a '&' O_el(Y)) 'or' (a '&' I_el(Y))).x by BVFUNC_1:2
.=(('not' a '&' O_el(Y)) 'or' a).x by BVFUNC_1:6
.=(O_el(Y) 'or' a).x by BVFUNC_1:5
.=a.x by BVFUNC_1:9;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a 'xor' b = 'not' a
'xor' 'not' b
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
('not' a 'xor' 'not' b).x =(('not' 'not' a '&' 'not' b) 'or' ('not' a
'&' 'not' 'not' b)).x by BVFUNC_4:9
.=(a 'xor' b).x by BVFUNC_4:9;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not'( a 'xor' b) = a
'xor' 'not' b
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
('not'( a 'xor' b)).x =('not'( ('not' a '&' b) 'or' (a '&' 'not' b))).
x by BVFUNC_4:9
.=(('not'('not' a '&' b) '&' 'not'( a '&' 'not' b))).x by BVFUNC_1:13
.=((('not' 'not' a 'or' 'not' b) '&' 'not'( a '&' 'not' b))).x by
BVFUNC_1:14
.=(((a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b))).x by BVFUNC_1:14
.=(((a 'or' 'not' b) '&' 'not' a 'or' (a 'or' 'not' b) '&' b)).x by
BVFUNC_1:12
.=(((a '&' 'not' a 'or' 'not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&'
b)).x by BVFUNC_1:12
.=(((O_el(Y) 'or' 'not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&' b)).x
by BVFUNC_4:5
.=((('not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&' b)).x by BVFUNC_1:9
.=((('not' b '&' 'not' a) 'or' (a '&' b 'or' 'not' b '&' b))).x by
BVFUNC_1:12
.=((('not' b '&' 'not' a) 'or' (a '&' b 'or' O_el(Y)))).x by BVFUNC_4:5
.=(('not' a '&' 'not' b) 'or' (a '&' 'not' 'not' b)).x by BVFUNC_1:9
.=(a 'xor' 'not' b).x by BVFUNC_4:9;
hence thesis;
end;
theorem Th18:
for a,b being Function of Y,BOOLEAN holds a 'eqv' b = (a
'or' 'not' b) '&' ('not' a 'or' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
((a 'or' 'not' b) '&' ('not' a 'or' b)).x =((a 'or' 'not' b) '&' (a
'imp' b)).x by BVFUNC_4:8
.=((a 'imp' b) '&' (b 'imp' a)).x by BVFUNC_4:8
.=(a 'eqv' b).x by BVFUNC_4:7;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a 'eqv' b = (a '&' b)
'or' ('not' a '&' 'not' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
((a '&' b) 'or' ('not' a '&' 'not' b)).x =(((a '&' b) 'or' 'not' a)
'&' ((a '&' b) 'or' 'not' b)).x by BVFUNC_1:11
.=(((a 'or' 'not' a) '&' (b 'or' 'not' a)) '&' ((a '&' b) 'or' 'not' b
)).x by BVFUNC_1:11
.=(((a 'or' 'not' a) '&' (b 'or' 'not' a)) '&' ((a 'or' 'not' b) '&' (
b 'or' 'not' b))).x by BVFUNC_1:11
.=((I_el(Y) '&' (b 'or' 'not' a)) '&' ((a 'or' 'not' b) '&' (b 'or'
'not' b))).x by BVFUNC_4:6
.=((I_el(Y) '&' (b 'or' 'not' a)) '&' ((a 'or' 'not' b) '&' I_el(Y))).
x by BVFUNC_4:6
.=((b 'or' 'not' a) '&' ((a 'or' 'not' b) '&' I_el(Y))).x by BVFUNC_1:6
.=((b 'or' 'not' a) '&' (a 'or' 'not' b)).x by BVFUNC_1:6
.=(a 'eqv' b).x by Th18;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'eqv' I_el(Y) = a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'eqv' I_el(Y)).x =((a 'imp' I_el(Y)) '&' (I_el(Y) 'imp' a)).x by
BVFUNC_4:7
.=(('not' a 'or' I_el(Y)) '&' (I_el(Y) 'imp' a)).x by BVFUNC_4:8
.=(('not' a 'or' I_el(Y)) '&' ('not' I_el(Y) 'or' a)).x by BVFUNC_4:8
.=(I_el(Y) '&' ('not' I_el(Y) 'or' a)).x by BVFUNC_1:10
.=(I_el(Y) '&' (O_el(Y) 'or' a)).x by BVFUNC_1:2
.=(I_el(Y) '&' a).x by BVFUNC_1:9
.=a.x by BVFUNC_1:6;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'eqv' O_el(Y) = 'not' a
proof
let a be Function of Y,BOOLEAN;
let x be Element of Y;
(a 'eqv' O_el(Y)).x =((a 'imp' O_el(Y)) '&' (O_el(Y) 'imp' a)).x by
BVFUNC_4:7
.=(('not' a 'or' O_el(Y)) '&' (O_el(Y) 'imp' a)).x by BVFUNC_4:8
.=(('not' a 'or' O_el(Y)) '&' ('not' O_el(Y) 'or' a)).x by BVFUNC_4:8
.=('not' a '&' ('not' O_el(Y) 'or' a)).x by BVFUNC_1:9
.=('not' a '&' (I_el(Y) 'or' a)).x by BVFUNC_1:2
.=('not' a '&' I_el(Y)).x by BVFUNC_1:10
.=('not' a).x by BVFUNC_1:6;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not'( a 'eqv' b) = (a
'eqv' 'not' b)
proof
let a,b be Function of Y,BOOLEAN;
let x be Element of Y;
('not'( a 'eqv' b)).x =('not'( (a 'imp' b) '&' (b 'imp' a))).x by
BVFUNC_4:7
.=('not'( ('not' a 'or' b) '&' (b 'imp' a))).x by BVFUNC_4:8
.=('not'( ('not' a 'or' b) '&' ('not' b 'or' a))).x by BVFUNC_4:8
.=('not'('not' a 'or' b) 'or' 'not'('not' b 'or' a)).x by BVFUNC_1:14
.=(('not' 'not' a '&' 'not' b) 'or' 'not'('not' b 'or' a)).x by
BVFUNC_1:13
.=((a '&' 'not' b) 'or' ('not' 'not' b '&' 'not' a)).x by BVFUNC_1:13
.=(((a '&' 'not' b) 'or' b) '&' ((a '&' 'not' b) 'or' 'not' a)).x by
BVFUNC_1:11
.=(((a 'or' b) '&' ('not' b 'or' b)) '&' ((a '&' 'not' b) 'or' 'not' a
)).x by BVFUNC_1:11
.=(((a 'or' b) '&' ('not' b 'or' b)) '&' ((a 'or' 'not' a) '&' ('not'
b 'or' 'not' a))).x by BVFUNC_1:11
.=(((a 'or' b) '&' I_el(Y)) '&' ((a 'or' 'not' a) '&' ('not' b 'or'
'not' a))).x by BVFUNC_4:6
.=(((a 'or' b) '&' I_el(Y)) '&' (I_el(Y) '&' ('not' b 'or' 'not' a))).
x by BVFUNC_4:6
.=((a 'or' b) '&' (I_el(Y) '&' ('not' b 'or' 'not' a))).x by BVFUNC_1:6
.=(('not' a 'or' 'not' b) '&' ('not' 'not' b 'or' a)).x by BVFUNC_1:6
.=(('not' a 'or' 'not' b) '&' ('not' b 'imp' a)).x by BVFUNC_4:8
.=((a 'imp' 'not' b) '&' ('not' b 'imp' a)).x by BVFUNC_4:8
.=(a 'eqv' 'not' b).x by BVFUNC_4:7;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not' a '<' (a 'imp' b
) 'eqv' 'not' a
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
assume
A1: ('not' a).z=TRUE;
((a 'imp' b) 'eqv' 'not' a).z =(('not' a 'or' b) 'eqv' 'not' a).z by
BVFUNC_4:8
.=((('not' a 'or' b) 'imp' 'not' a) '&' ('not' a 'imp' ('not' a 'or' b))
).z by BVFUNC_4:7
.=(('not'('not' a 'or' b) 'or' 'not' a) '&' ('not' a 'imp' ('not' a 'or'
b))).z by BVFUNC_4:8
.=(('not'('not' a 'or' b) 'or' 'not' a) '&' ('not' 'not' a 'or' ('not' a
'or' b))).z by BVFUNC_4:8
.=('not'('not' a 'or' b) 'or' 'not' a).z '&' ('not' 'not' a 'or' ('not'
a 'or' b)).z by MARGREL1:def 20
.=(('not'('not' a 'or' b)).z 'or' ('not' a).z) '&' ('not' 'not' a 'or' (
'not' a 'or' b)).z by BVFUNC_1:def 4
.=('not' ('not' a 'or' b).z 'or' ('not' a).z) '&' ('not' 'not' a 'or' (
'not' a 'or' b)).z by MARGREL1:def 19
.=('not'( ('not' a).z 'or' b.z) 'or' ('not' a).z) '&' ('not' 'not' a
'or' ('not' a 'or' b)).z by BVFUNC_1:def 4
.=(('not' 'not' a.z '&' 'not' b.z) 'or' ('not' a).z) '&' ('not'
'not' a 'or' ('not' a 'or' b)).z by MARGREL1:def 19
.=((a.z '&' 'not' b.z) 'or' ('not' a).z) '&' (('not' 'not' a).z 'or'
('not' a 'or' b).z) by BVFUNC_1:def 4
.=((a.z '&' 'not' b.z) 'or' ('not' a).z) '&' (a.z 'or' (('not' a).
z 'or' b.z)) by BVFUNC_1:def 4
.=TRUE '&' (FALSE 'or' (TRUE 'or' b.z)) by A1
.=FALSE 'or' (TRUE 'or' b.z)
.=(TRUE 'or' b.z)
.=TRUE;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds 'not' a '<' (b 'imp' a
) 'eqv' 'not' b
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
assume ('not' a).z=TRUE;
then
A1: 'not' a.z=TRUE by MARGREL1:def 19;
((b 'imp' a) 'eqv' 'not' b).z =(('not' b 'or' a) 'eqv' 'not' b).z by
BVFUNC_4:8
.=((('not' b 'or' a) 'imp' 'not' b) '&' ('not' b 'imp' ('not' b 'or' a))
).z by BVFUNC_4:7
.=(('not'('not' b 'or' a) 'or' 'not' b) '&' ('not' b 'imp' ('not' b 'or'
a))).z by BVFUNC_4:8
.=(('not'('not' b 'or' a) 'or' 'not' b) '&' ('not' 'not' b 'or' ('not' b
'or' a))).z by BVFUNC_4:8
.=('not'('not' b 'or' a) 'or' 'not' b).z '&' ('not' 'not' b 'or' ('not'
b 'or' a)).z by MARGREL1:def 20
.=(('not'('not' b 'or' a)).z 'or' ('not' b).z) '&' ('not' 'not' b 'or' (
'not' b 'or' a)).z by BVFUNC_1:def 4
.=('not' ('not' b 'or' a).z 'or' ('not' b).z) '&' ('not' 'not' b 'or' (
'not' b 'or' a)).z by MARGREL1:def 19
.=('not'( ('not' b).z 'or' a.z) 'or' ('not' b).z) '&' ('not' 'not' b
'or' ('not' b 'or' a)).z by BVFUNC_1:def 4
.=(('not' 'not' b.z '&' 'not' a.z) 'or' ('not' b).z) '&' ('not'
'not' b 'or' ('not' b 'or' a)).z by MARGREL1:def 19
.=((b.z '&' 'not' a.z) 'or' ('not' b).z) '&' (('not' 'not' b).z 'or'
('not' b 'or' a).z) by BVFUNC_1:def 4
.=((b.z '&' 'not' a.z) 'or' ('not' b).z) '&' (('not' 'not' b).z 'or'
(('not' b).z 'or' a.z)) by BVFUNC_1:def 4
.=((b.z '&' 'not' a.z) 'or' ('not' b).z) '&' (b.z 'or' ('not' b.
z 'or' a.z)) by MARGREL1:def 19
.=((TRUE '&' b.z) 'or' ('not' b).z) '&' (b.z 'or' ('not' b.z 'or'
FALSE)) by A1
.=(b.z 'or' ('not' b).z) '&' (b.z 'or' ('not' b.z 'or' FALSE))
.=(b.z 'or' 'not' b.z) '&' (b.z 'or' ('not' b.z 'or' FALSE)) by
MARGREL1:def 19
.=TRUE '&' (b.z 'or' ('not' b.z 'or' FALSE))
.=b.z 'or' ('not' b.z 'or' FALSE)
.=b.z 'or' 'not' b.z 'or' FALSE
.=TRUE 'or' FALSE by XBOOLEAN:102
.=TRUE;
hence thesis;
end;
theorem
for a,b being Function of Y,BOOLEAN holds a '<' (a 'or' b) 'eqv'
(b 'or' a) 'eqv' a
proof
let a,b be Function of Y,BOOLEAN;
let z be Element of Y;
assume
A1: a.z=TRUE;
A2: ((a 'or' b) 'eqv' (b 'or' a)).z =(((a 'or' b) 'imp' (a 'or' b)) '&' ((a
'or' b) 'imp' (a 'or' b))).z by BVFUNC_4:7
.=('not'( a 'or' b) 'or' (a 'or' b)).z by BVFUNC_4:8
.=(I_el(Y)).z by BVFUNC_4:6
.=TRUE by BVFUNC_1:def 11;
((a 'or' b) 'eqv' (b 'or' a) 'eqv' a).z =((((a 'or' b) 'eqv' (a 'or' b))
'imp' a) '&' (a 'imp' ((a 'or' b) 'eqv' (a 'or' b)))).z by BVFUNC_4:7
.=(((a 'or' b) 'eqv' (a 'or' b)) 'imp' a).z '&' (a 'imp' ((a 'or' b)
'eqv' (a 'or' b))).z by MARGREL1:def 20
.=('not'( (a 'or' b) 'eqv' (a 'or' b)) 'or' a).z '&' (a 'imp' ((a 'or' b
) 'eqv' (a 'or' b))).z by BVFUNC_4:8
.=('not'( (a 'or' b) 'eqv' (a 'or' b)) 'or' a).z '&' ('not' a 'or' ((a
'or' b) 'eqv' (a 'or' b))).z by BVFUNC_4:8
.=(('not'( (a 'or' b) 'eqv' (a 'or' b))).z 'or' a.z) '&' ('not' a 'or'
((a 'or' b) 'eqv' (a 'or' b))).z by BVFUNC_1:def 4
.=(('not'( (a 'or' b) 'eqv' (a 'or' b))).z 'or' a.z) '&' (('not' a).z
'or' ((a 'or' b) 'eqv' (a 'or' b)).z) by BVFUNC_1:def 4
.=('not' ((a 'or' b) 'eqv' (a 'or' b)).z 'or' a.z) '&' (('not' a).z
'or' ((a 'or' b) 'eqv' (a 'or' b)).z) by MARGREL1:def 19
.=(FALSE 'or' a.z) '&' (('not' a).z 'or' TRUE) by A2
.=a.z '&' (('not' a).z 'or' TRUE)
.=a.z '&' TRUE
.=TRUE by A1;
hence thesis;
end;
theorem
for a being Function of Y,BOOLEAN holds a 'imp' ('not' a 'eqv'
'not' a) = I_el(Y)
proof
let a be Function of Y,BOOLEAN;
for x being Element of Y holds (a 'imp' ('not' a 'eqv' 'not' a)).x = TRUE
proof
let x be Element of Y;
(a 'imp' ('not' a 'eqv' 'not' a)).x =('not' a 'or' ('not' a 'eqv'
'not' a)).x by BVFUNC_4:8
.=('not' a 'or' (('not' a 'imp' 'not' a) '&' ('not' a 'imp' 'not' a)))
.x by BVFUNC_4:7
.=('not' a 'or' ('not' 'not' a 'or' 'not' a)).x by BVFUNC_4:8
.=('not' a 'or' I_el(Y)).x by BVFUNC_4:6
.=TRUE by BVFUNC_1:10,def 11;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b being Function of Y,BOOLEAN holds ((a 'imp' b) 'imp' a)
'imp' a = I_el(Y)
proof
let a,b be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'imp' b) 'imp' a) 'imp' a).x = TRUE
proof
let x be Element of Y;
(((a 'imp' b) 'imp' a) 'imp' a).x =('not'( (a 'imp' b) 'imp' a) 'or' a
).x by BVFUNC_4:8
.=('not'('not'( a 'imp' b) 'or' a) 'or' a).x by BVFUNC_4:8
.=('not'('not'('not' a 'or' b) 'or' a) 'or' a).x by BVFUNC_4:8
.=('not'( ('not' 'not' a '&' 'not' b) 'or' a) 'or' a).x by BVFUNC_1:13
.=('not'( ((a 'or' a) '&' ('not' b 'or' a))) 'or' a).x by BVFUNC_1:11
.=(('not' a 'or' 'not'('not' b 'or' a)) 'or' a).x by BVFUNC_1:14
.=(('not' a 'or' ('not' 'not' b '&' 'not' a)) 'or' a).x by BVFUNC_1:13
.=((('not' a 'or' b) '&' ('not' a 'or' 'not' a)) 'or' a).x by BVFUNC_1:11
.=((('not' a 'or' b) 'or' a) '&' ('not' a 'or' a)).x by BVFUNC_1:11
.=((('not' a 'or' b) 'or' a) '&' I_el(Y)).x by BVFUNC_4:6
.=(('not' a 'or' b) 'or' a).x by BVFUNC_1:6
.=(b 'or' ('not' a 'or' a)).x by BVFUNC_1:8
.=(b 'or' I_el(Y)).x by BVFUNC_4:6
.=TRUE by BVFUNC_1:10,def 11;
hence thesis;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds ((a 'imp' c) '&' (
b 'imp' d)) '&' ('not' c 'or' 'not' d) 'imp' ('not' a 'or' 'not' b)=I_el(Y)
proof
let a,b,c,d be Function of Y,BOOLEAN;
for x being Element of Y holds (((a 'imp' c) '&' (b 'imp' d)) '&' ('not'
c 'or' 'not' d) 'imp' ('not' a 'or' 'not' b)).x = TRUE
proof
let x be Element of Y;
((a 'imp' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d) 'imp' ('not'
a 'or' 'not' b) ='not'( ((a 'imp' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d
)) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:8
.='not'( (('not' a 'or' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d)
) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:8
.='not'( (('not' a 'or' c) '&' ('not' b 'or' d)) '&' ('not' c 'or'
'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:8
.=('not'( ('not' a 'or' c) '&' ('not' b 'or' d)) 'or' 'not'('not' c
'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:14
.=(('not'('not' a 'or' c) 'or' 'not'('not' b 'or' d)) 'or' 'not'('not'
c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:14
.=((('not' 'not' a '&' 'not' c) 'or' 'not'('not' b 'or' d)) 'or' 'not'
('not' c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:13
.=(((a '&' 'not' c) 'or' ('not' 'not' b '&' 'not' d)) 'or' 'not'('not'
c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:13
.=(((a '&' 'not' c) 'or' (b '&' 'not' d)) 'or' ('not' 'not' c '&'
'not' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:13
.=((a '&' 'not' c) 'or' ((b '&' 'not' d) 'or' (c '&' d))) 'or' ('not'
a 'or' 'not' b) by BVFUNC_1:8
.=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' ('not' d 'or' (c '&' d
)))) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:11
.=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' (('not' d 'or' c) '&'
('not' d 'or' d)))) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:11
.=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' (('not' d 'or' c) '&'
I_el(Y)))) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:6
.=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' ('not' d 'or' c)))
'or' ('not' a 'or' 'not' b) by BVFUNC_1:6
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ((a '&' 'not' c) 'or'
('not' a 'or' 'not' b)) by BVFUNC_1:8
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ((a 'or' ('not' a
'or' 'not' b)) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:11
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' (((a 'or' 'not' a)
'or' 'not' b) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:8
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ((I_el(Y) 'or' 'not'
b) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_4:6
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' (I_el(Y) '&' ('not' c
'or' ('not' a 'or' 'not' b))) by BVFUNC_1:10
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ('not' c 'or' ('not'
a 'or' 'not' b)) by BVFUNC_1:6
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
(('not' d 'or' c) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:11
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
((('not' d 'or' c) 'or' 'not' c) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:8
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
(('not' d 'or' (c 'or' 'not' c)) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:8
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
(('not' d 'or' I_el(Y)) 'or' ('not' a 'or' 'not' b)) by BVFUNC_4:6
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
(I_el(Y) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:10
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
I_el(Y) by BVFUNC_1:10
.=(b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b)) by
BVFUNC_1:6
.=(c '&' d) 'or' (b 'or' ('not' c 'or' ('not' a 'or' 'not' b))) by
BVFUNC_1:8
.=(c '&' d) 'or' ((b 'or' ('not' b 'or' 'not' a)) 'or' 'not' c) by
BVFUNC_1:8
.=(c '&' d) 'or' (((b 'or' 'not' b) 'or' 'not' a) 'or' 'not' c) by
BVFUNC_1:8
.=(c '&' d) 'or' ((I_el(Y) 'or' 'not' a) 'or' 'not' c) by BVFUNC_4:6
.=(c '&' d) 'or' (I_el(Y) 'or' 'not' c) by BVFUNC_1:10
.=(c '&' d) 'or' I_el(Y) by BVFUNC_1:10
.=I_el(Y) by BVFUNC_1:10;
hence thesis by BVFUNC_1:def 11;
end;
hence thesis by BVFUNC_1:def 11;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((
a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
for x being Element of Y holds ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c))
'imp' (a 'imp' c))).x=TRUE
proof
let x be Element of Y;
(a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) ='not' (a
'imp' b) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8
.='not'('not' a 'or' b) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))
by BVFUNC_4:8
.='not'('not' a 'or' b) 'or' (('not' a 'or' (b 'imp' c)) 'imp' (a
'imp' c)) by BVFUNC_4:8
.='not'('not' a 'or' b) 'or' (('not' a 'or' ('not' b 'or' c)) 'imp' (a
'imp' c)) by BVFUNC_4:8
.='not'('not' a 'or' b) 'or' (('not' a 'or' ('not' b 'or' c)) 'imp' (
'not' a 'or' c)) by BVFUNC_4:8
.='not'('not' a 'or' b) 'or' ('not'('not' a 'or' ('not' b 'or' c))
'or' ('not' a 'or' c)) by BVFUNC_4:8
.=('not' 'not' a '&' 'not' b) 'or' ('not'('not' a 'or' ('not' b 'or' c
)) 'or' ( 'not' a 'or' c)) by BVFUNC_1:13
.=('not' 'not' a '&' 'not' b) 'or' (('not' 'not' a '&' 'not'('not' b
'or' c)) 'or' ('not' a 'or' c)) by BVFUNC_1:13
.=(a '&' 'not' b) 'or' (('not' 'not' a '&' ('not' 'not' b '&' 'not' c)
) 'or' ('not' a 'or' c)) by BVFUNC_1:13
.=(a '&' 'not' b) 'or' ((a 'or' ('not' a 'or' c)) '&' ((b '&' 'not' c)
'or' ('not' a 'or' c))) by BVFUNC_1:11
.=(a '&' 'not' b) 'or' (((a 'or' 'not' a) 'or' c) '&' ((b '&' 'not' c)
'or' ('not' a 'or' c))) by BVFUNC_1:8
.=(a '&' 'not' b) 'or' ((I_el(Y) 'or' c) '&' ((b '&' 'not' c) 'or' (
'not' a 'or' c))) by BVFUNC_4:6
.=(a '&' 'not' b) 'or' (I_el(Y) '&' ((b '&' 'not' c) 'or' ('not' a
'or' c))) by BVFUNC_1:10
.=(a '&' 'not' b) 'or' ((b '&' 'not' c) 'or' ('not' a 'or' c)) by
BVFUNC_1:6
.=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' ('not' c 'or' (
'not' a 'or' c))) by BVFUNC_1:11
.=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' (('not' c 'or' c
) 'or' 'not' a)) by BVFUNC_1:8
.=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' (I_el(Y) 'or'
'not' a)) by BVFUNC_4:6
.=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' I_el(Y)) by
BVFUNC_1:10
.=(a '&' 'not' b) 'or' (b 'or' ('not' a 'or' c)) by BVFUNC_1:6
.=(a 'or' (b 'or' ('not' a 'or' c))) '&' ('not' b 'or' (b 'or' ('not'
a 'or' c))) by BVFUNC_1:11
.=(a 'or' (b 'or' ('not' a 'or' c))) '&' (('not' b 'or' b) 'or' ('not'
a 'or' c)) by BVFUNC_1:8
.=(a 'or' (b 'or' ('not' a 'or' c))) '&' (I_el(Y) 'or' ('not' a 'or' c
)) by BVFUNC_4:6
.=(a 'or' (b 'or' ('not' a 'or' c))) '&' I_el(Y) by BVFUNC_1:10
.=a 'or' (b 'or' ('not' a 'or' c)) by BVFUNC_1:6
.=a 'or' (('not' a 'or' b) 'or' c) by BVFUNC_1:8
.=a 'or' ('not' a 'or' b) 'or' c by BVFUNC_1:8
.=(a 'or' 'not' a) 'or' b 'or' c by BVFUNC_1:8
.=I_el(Y) 'or' b 'or' c by BVFUNC_4:6
.=I_el(Y) 'or' c by BVFUNC_1:10
.=I_el(Y) by BVFUNC_1:10;
hence thesis by BVFUNC_1:def 11;
end;
hence thesis by BVFUNC_1:def 11;
end;
begin :: BVFUNC_9
reserve Y for non empty set,
a,b,c,d,e,f,g for Function of Y,BOOLEAN;
Lm1: a '&' b '<' a
proof
let x be Element of Y;
assume (a '&' b).x = TRUE;
then a.x '&' b.x = TRUE by MARGREL1:def 20;
hence thesis by MARGREL1:12;
end;
Lm2: a '&' b '&' c '<' a & a '&' b '&' c '<' b
proof
a '&' b '&' c = c '&' b '&' a & c '&' b '&' a 'imp' a = I_el(Y) by BVFUNC_1:4
,Th38;
hence a '&' b '&' c '<' a by BVFUNC_1:16;
a '&' b '&' c = a '&' c '&' b & a '&' c '&' b 'imp' b = I_el(Y) by BVFUNC_1:4
,Th38;
hence thesis by BVFUNC_1:16;
end;
Lm3: a '&' b '&' c '&' d '<' a & a '&' b '&' c '&' d '<' b
proof
A1: d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38;
a '&' b '&' c '&' d = d '&' c '&' (b '&' a) by BVFUNC_1:4
.=d '&' c '&' b '&' a by BVFUNC_1:4;
hence a '&' b '&' c '&' d '<' a by A1,BVFUNC_1:16;
A2: a '&' c '&' d '&' b 'imp' b = I_el(Y) by Th38;
a '&' b '&' c '&' d = a '&' c '&' b '&' d by BVFUNC_1:4
.=a '&' c '&' d '&' b by BVFUNC_1:4;
hence thesis by A2,BVFUNC_1:16;
end;
Lm4: a '&' b '&' c '&' d '&' e '<' a & a '&' b '&' c '&' d '&' e '<' b
proof
A1: e '&' d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38;
a '&' b '&' c '&' d '&' e =e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4
.=e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4
.=e '&' d '&' c '&' b '&' a by BVFUNC_1:4;
hence a '&' b '&' c '&' d '&' e '<' a by A1,BVFUNC_1:16;
A2: a '&' c '&' d '&' e '&' b 'imp' b = I_el(Y) by Th38;
a '&' b '&' c '&' d '&' e =a '&' c '&' b '&' d '&' e by BVFUNC_1:4
.=a '&' c '&' d '&' b '&' e by BVFUNC_1:4
.=a '&' c '&' d '&' e '&' b by BVFUNC_1:4;
hence thesis by A2,BVFUNC_1:16;
end;
Lm5: a '&' b '&' c '&' d '&' e '&' f '<' a & a '&' b '&' c '&' d '&' e '&' f
'<' b
proof
A1: f '&' e '&' d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38;
a '&' b '&' c '&' d '&' e '&' f =f '&' e '&' (d '&' (c '&' (b '&' a)))
by BVFUNC_1:4
.=f '&' e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4
.=f '&' e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4
.=f '&' e '&' d '&' c '&' b '&' a by BVFUNC_1:4;
hence a '&' b '&' c '&' d '&' e '&' f '<' a by A1,BVFUNC_1:16;
A2: f '&' e '&' d '&' c '&' a '&' b 'imp' b = I_el(Y) by Th38;
a '&' b '&' c '&' d '&' e '&' f =f '&' e '&' (d '&' (c '&' (b '&' a)))
by BVFUNC_1:4
.=f '&' e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4
.=f '&' e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4
.=f '&' e '&' d '&' c '&' a '&' b by BVFUNC_1:4;
hence thesis by A2,BVFUNC_1:16;
end;
Lm6: a '&' b '&' c '&' d '&' e '&' f '&' g '<' a & a '&' b '&' c '&' d '&' e
'&' f '&' g '<' b
proof
A1: g '&' f '&' e '&' d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38;
a '&' b '&' c '&' d '&' e '&' f '&' g =g '&' f '&' (e '&' (d '&' (c '&'
(b '&' a)))) by BVFUNC_1:4
.=g '&' f '&' e '&' (d '&' (c '&' (b '&' a))) by BVFUNC_1:4
.=g '&' f '&' e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4
.=g '&' f '&' e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4
.=g '&' f '&' e '&' d '&' c '&' b '&' a by BVFUNC_1:4;
hence a '&' b '&' c '&' d '&' e '&' f '&' g '<' a by A1,BVFUNC_1:16;
A2: a '&' g '&' f '&' e '&' d '&' c '&' b 'imp' b = I_el(Y) by Th38;
a '&' b '&' c '&' d '&' e '&' f '&' g =a '&' (c '&' b) '&' d '&' e '&' f
'&' g by BVFUNC_1:4
.=a '&' (d '&' (c '&' b)) '&' e '&' f '&' g by BVFUNC_1:4
.=a '&' (e '&' (d '&' (c '&' b))) '&' f '&' g by BVFUNC_1:4
.=a '&' (f '&' (e '&' (d '&' (c '&' b)))) '&' g by BVFUNC_1:4
.=a '&' g '&' (f '&' (e '&' (d '&' (c '&' b)))) by BVFUNC_1:4
.=a '&' g '&' f '&' (e '&' (d '&' (c '&' b))) by BVFUNC_1:4
.=a '&' g '&' f '&' e '&' (d '&' (c '&' b)) by BVFUNC_1:4
.=a '&' g '&' f '&' e '&' d '&' (c '&' b) by BVFUNC_1:4
.=a '&' g '&' f '&' e '&' d '&' c '&' b by BVFUNC_1:4;
hence thesis by A2,BVFUNC_1:16;
end;
theorem Th1:
(a 'or' b) '&' (b 'imp' c) '<' a 'or' c
proof
let z be Element of Y;
A1: ((a 'or' b) '&' (b 'imp' c)).z =(a 'or' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=(a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(a 'or' b).z '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'or' b) '&' (b 'imp' c)).z=TRUE;
now
assume (a 'or' c).z<>TRUE;
then (a 'or' c).z=FALSE by XBOOLEAN:def 3;
then
A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4;
c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
then (a.z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&' 'not' b.z by A3,
MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem Th2:
a '&' (a 'imp' b) '<' b
proof
let z be Element of Y;
A1: (a '&' (a 'imp' b)).z =a.z '&' (a 'imp' b).z by MARGREL1:def 20
.=a.z '&' ('not' a 'or' b).z by BVFUNC_4:8
.=a.z '&' (('not' a).z 'or' b.z) by BVFUNC_1:def 4
.=a.z '&' ('not' a).z 'or' a.z '&' b.z by XBOOLEAN:8
.=a.z '&' 'not' a.z 'or' a.z '&' b.z by MARGREL1:def 19
.=FALSE 'or' a.z '&' b.z by XBOOLEAN:138
.=a.z '&' b.z;
assume
A2: (a '&' (a 'imp' b)).z=TRUE;
now
assume b.z<>TRUE;
then a.z '&' b.z = FALSE '&' a.z by XBOOLEAN:def 3
.= FALSE;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'imp' b) '&' 'not' b '<' 'not' a
proof
let z be Element of Y;
reconsider bz = b.z as boolean object;
A1: ((a 'imp' b) '&' 'not' b).z =(a 'imp' b).z '&' ('not' b).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' ('not' b).z by BVFUNC_4:8
.=('not' b).z '&' (('not' a).z 'or' b.z) by BVFUNC_1:def 4
.=('not' b).z '&' ('not' a).z 'or' ('not' b).z '&' b.z by XBOOLEAN:8
.=('not' b).z '&' ('not' a).z 'or' 'not' bz '&' bz by MARGREL1:def 19
.=('not' b).z '&' ('not' a).z 'or' FALSE by XBOOLEAN:138
.=('not' b).z '&' ('not' a).z;
assume
A2: ((a 'imp' b) '&' 'not' b).z=TRUE;
now
assume ('not' a).z<>TRUE;
then ('not' b).z '&' ('not' a).z =FALSE '&' ('not' b).z by XBOOLEAN:def 3
.=FALSE;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'or' b) '&' 'not' a '<' b
proof
let z be Element of Y;
reconsider az = a.z as boolean object;
A1: ((a 'or' b) '&' 'not' a).z =(a 'or' b).z '&' ('not' a).z by MARGREL1:def 20
.=('not' a).z '&' (a.z 'or' b.z) by BVFUNC_1:def 4
.=('not' a).z '&' a.z 'or' ('not' a).z '&' b.z by XBOOLEAN:8
.='not' az '&' az 'or' ('not' a).z '&' b.z by MARGREL1:def 19
.=FALSE 'or' ('not' a).z '&' b.z by XBOOLEAN:138
.=('not' a).z '&' b.z;
assume
A2: ((a 'or' b) '&' 'not' a).z=TRUE;
now
assume b.z<>TRUE;
then ('not' a).z '&' b.z =FALSE '&' ('not' a).z by XBOOLEAN:def 3
.=FALSE;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'imp' b) '&' ('not' a 'imp' b) '<' b
proof
let z be Element of Y;
reconsider az = a.z as boolean object;
A1: ((a 'imp' b) '&' ('not' a 'imp' b)).z =(a 'imp' b).z '&' ('not' a 'imp'
b).z by MARGREL1:def 20
.=('not' a 'or' b).z '&' ('not' a 'imp' b).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' 'not' a 'or' b).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' (a 'or' b).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (a.z 'or' b.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' ('not' a 'imp' b)).z=TRUE;
now
assume b.z<>TRUE;
then b.z=FALSE by XBOOLEAN:def 3;
then (('not' a).z 'or' b.z) '&' (a.z 'or' b.z) ='not' az '&' az by
MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'imp' b) '&' (a 'imp' 'not' b) '<' 'not' a
proof
let z be Element of Y;
A1: ((a 'imp' b) '&' (a 'imp' 'not' b)).z =(a 'imp' b).z '&' (a 'imp' 'not'
b).z by MARGREL1:def 20
.=('not' a 'or' b).z '&' (a 'imp' 'not' b).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' a 'or' 'not' b).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' a 'or' 'not' b).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' a).z 'or' ('not' b).z) by
BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (a 'imp' 'not' b)).z=TRUE;
now
assume ('not' a).z<>TRUE;
then ('not' a).z=FALSE by XBOOLEAN:def 3;
then (('not' a).z 'or' b.z) '&' (('not' a).z 'or' ('not' b).z) =b.z '&'
'not' b.z by MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
a 'imp' (b '&' c) '<' a 'imp' b
proof
let z be Element of Y;
A1: (a 'imp' (b '&' c)).z =('not' a 'or' (b '&' c)).z by BVFUNC_4:8
.=('not' a).z 'or' (b '&' c).z by BVFUNC_1:def 4
.=('not' a).z 'or' (b.z '&' c.z) by MARGREL1:def 20;
assume
A2: (a 'imp' (b '&' c)).z=TRUE;
now
assume (a 'imp' b).z<>TRUE;
then (a 'imp' b).z=FALSE by XBOOLEAN:def 3;
then ('not' a 'or' b).z=FALSE by BVFUNC_4:8;
then
A3: ('not' a).z 'or' b.z=FALSE by BVFUNC_1:def 4;
('not' a).z 'or' (b.z '&' c.z) =(('not' a).z 'or' b.z) '&' (('not' a).
z 'or' c.z) by XBOOLEAN:9
.=FALSE by A3;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'or' b) 'imp' c '<' a 'imp' c
proof
let z be Element of Y;
A1: ((a 'or' b) 'imp' c).z =('not'( a 'or' b) 'or' c).z by BVFUNC_4:8
.=(('not' a '&' 'not' b) 'or' c).z by BVFUNC_1:13
.=('not' a '&' 'not' b).z 'or' c.z by BVFUNC_1:def 4
.=(('not' a).z '&' ('not' b).z) 'or' c.z by MARGREL1:def 20;
assume
A2: ((a 'or' b) 'imp' c).z=TRUE;
now
assume (a 'imp' c).z<>TRUE;
then (a 'imp' c).z=FALSE by XBOOLEAN:def 3;
then ('not' a 'or' c).z=FALSE by BVFUNC_4:8;
then
A3: ('not' a).z 'or' c.z=FALSE by BVFUNC_1:def 4;
(('not' a).z '&' ('not' b).z) 'or' c.z =(c.z 'or' ('not' a).z) '&' (c.
z 'or' ('not' b).z) by XBOOLEAN:9
.=FALSE by A3;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
a 'imp' b '<' (a '&' c) 'imp' b
proof
let z be Element of Y;
A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8
.=('not' a).z 'or' b.z by BVFUNC_1:def 4;
assume
A2: (a 'imp' b).z=TRUE;
now
assume ((a '&' c) 'imp' b).z<>TRUE;
then ((a '&' c) 'imp' b).z=FALSE by XBOOLEAN:def 3;
then ('not'( a '&' c) 'or' b).z=FALSE by BVFUNC_4:8;
then ('not'( a '&' c)).z 'or' b.z=FALSE by BVFUNC_1:def 4;
then ('not' a 'or' 'not' c) .z 'or' b.z=FALSE by BVFUNC_1:14;
then (('not' c).z 'or' ('not' a).z) 'or' b.z=FALSE by BVFUNC_1:def 4;
then ('not' c).z 'or' (('not' a).z 'or' b.z)=FALSE;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
a 'imp' b '<' (a '&' c) 'imp' (b '&' c)
proof
let z be Element of Y;
A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8
.=('not' a).z 'or' b.z by BVFUNC_1:def 4;
assume
A2: (a 'imp' b).z=TRUE;
now
assume ((a '&' c) 'imp' (b '&' c)).z<>TRUE;
then ((a '&' c) 'imp' (b '&' c)).z=FALSE by XBOOLEAN:def 3;
then ('not'( a '&' c) 'or' (b '&' c)).z=FALSE by BVFUNC_4:8;
then ('not'( a '&' c)).z 'or' (b '&' c).z=FALSE by BVFUNC_1:def 4;
then ('not' a 'or' 'not' c) .z 'or' (b '&' c).z=FALSE by BVFUNC_1:14;
then (('not' c).z 'or' ('not' a).z) 'or' (b '&' c).z=FALSE by
BVFUNC_1:def 4;
then ('not' c).z 'or' (('not' a).z 'or' (b '&' c).z)=FALSE;
then
A3: ('not' c).z 'or' (('not' a).z 'or' (b.z '&' c.z))=FALSE by MARGREL1:def 20;
('not' c).z 'or' ((('not' a).z 'or' b.z) '&' (('not' a).z 'or' c.z))
=('not' c).z 'or' c.z 'or' ('not' a).z by A2,A1
.='not' c.z 'or' c.z 'or' ('not' a).z by MARGREL1:def 19
.=TRUE 'or' ('not' a).z by XBOOLEAN:102
.=TRUE;
hence contradiction by A3,XBOOLEAN:9;
end;
hence thesis;
end;
theorem
a 'imp' b '<' a 'imp' (b 'or' c)
proof
let z be Element of Y;
A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8
.=('not' a).z 'or' b.z by BVFUNC_1:def 4;
assume
A2: (a 'imp' b).z=TRUE;
now
assume
A3: (a 'imp' (b 'or' c)).z<>TRUE;
(a 'imp' (b 'or' c)).z =('not' a 'or' (b 'or' c)).z by BVFUNC_4:8
.=('not' a).z 'or' (b 'or' c).z by BVFUNC_1:def 4
.=('not' a).z 'or' (b.z 'or' c.z) by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) 'or' c.z
.=TRUE by A2,A1;
hence contradiction by A3;
end;
hence thesis;
end;
theorem
a 'imp' b '<' (a 'or' c) 'imp' (b 'or' c)
proof
let z be Element of Y;
A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8
.=('not' a).z 'or' b.z by BVFUNC_1:def 4;
assume
A2: (a 'imp' b).z=TRUE;
now
assume
A3: ((a 'or' c) 'imp' (b 'or' c)).z<>TRUE;
((a 'or' c) 'imp' (b 'or' c)).z =('not'( a 'or' c) 'or' (b 'or' c)).z
by BVFUNC_4:8
.=('not'( a 'or' c)).z 'or' (b 'or' c).z by BVFUNC_1:def 4
.=('not'( a 'or' c)).z 'or' (b.z 'or' c.z) by BVFUNC_1:def 4
.=(('not'( a 'or' c)).z 'or' b.z) 'or' c.z
.=((('not' a '&' 'not' c)).z 'or' b.z) 'or' c.z by BVFUNC_1:13
.=(b.z 'or' (('not' a).z '&' ('not' c).z)) 'or' c.z by MARGREL1:def 20
.=((('not' a).z 'or' b.z) '&' (b.z 'or' ('not' c).z)) 'or' c.z by
XBOOLEAN:9
.=b.z 'or' (('not' c).z 'or' c.z) by A2,A1
.=b.z 'or' ('not' c.z 'or' c.z) by MARGREL1:def 19
.=b.z 'or' TRUE by XBOOLEAN:102
.=TRUE;
hence contradiction by A3;
end;
hence thesis;
end;
theorem
a '&' b 'or' c '<' a 'or' c
proof
let z be Element of Y;
A1: (a '&' b 'or' c).z =(a '&' b).z 'or' c.z by BVFUNC_1:def 4
.=(a.z '&' b.z) 'or' c.z by MARGREL1:def 20;
assume
A2: (a '&' b 'or' c).z=TRUE;
now
assume (a 'or' c).z<>TRUE;
then (a 'or' c).z=FALSE by XBOOLEAN:def 3;
then
A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4;
(a.z '&' b.z) 'or' c.z =(c.z 'or' a.z) '&' (c.z 'or' b.z) by XBOOLEAN:9
.=FALSE by A3;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a '&' b) 'or' (c '&' d) '<' a 'or' c
proof
let z be Element of Y;
A1: ((a '&' b) 'or' (c '&' d)).z =(a '&' b).z 'or' (c '&' d).z by
BVFUNC_1:def 4
.=(a.z '&' b.z) 'or' (c '&' d).z by MARGREL1:def 20
.=(a.z '&' b.z) 'or' (c.z '&' (d).z) by MARGREL1:def 20;
assume
A2: ((a '&' b) 'or' (c '&' d)).z=TRUE;
now
assume (a 'or' c).z<>TRUE;
then (a 'or' c).z=FALSE by XBOOLEAN:def 3;
then
A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4;
(a.z '&' b.z) 'or' (c.z '&' (d).z) =(c.z 'or' (a.z '&' b.z)) '&' ((a.z
'&' b.z) 'or' (d).z) by XBOOLEAN:9
.=((a.z 'or' c.z) '&' (c.z 'or' b.z)) '&' ((a.z '&' b.z) 'or' (d).z)
by XBOOLEAN:9
.=FALSE by A3;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'imp' b) '&' ('not' a 'imp' c) '<' b 'or' c
proof
let z be Element of Y;
A1: ((a 'imp' b) '&' ('not' a 'imp' c)).z =(('not' a 'or' b) '&' ('not' a
'imp' c)).z by BVFUNC_4:8
.=(('not' a 'or' b) '&' ('not' 'not' a 'or' c)).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' (a 'or' c).z by MARGREL1:def 20
.=(('not' a).z 'or' b.z) '&' (a 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (a.z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' ('not' a 'imp' c)).z=TRUE;
now
reconsider az = a.z as boolean object;
assume (b 'or' c).z<>TRUE;
then (b 'or' c).z=FALSE by XBOOLEAN:def 3;
then
A3: b.z 'or' c.z=FALSE by BVFUNC_1:def 4;
c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
then (('not' a).z 'or' b.z) '&' (a.z 'or' c.z) ='not' az '&' az by A3,
MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'imp' c) '&' (b 'imp' 'not' c) '<' 'not' a 'or' 'not' b
proof
let z be Element of Y;
A1: ((a 'imp' c) '&' (b 'imp' 'not' c)).z =(a 'imp' c).z '&' (b 'imp' 'not'
c).z by MARGREL1:def 20
.=('not' a 'or' c).z '&' (b 'imp' 'not' c).z by BVFUNC_4:8
.=('not' a 'or' c).z '&' ('not' b 'or' 'not' c).z by BVFUNC_4:8
.=(('not' a).z 'or' c.z) '&' ('not' b 'or' 'not' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' c.z) '&' (('not' b).z 'or' ('not' c).z) by
BVFUNC_1:def 4;
assume
A2: ((a 'imp' c) '&' (b 'imp' 'not' c)).z=TRUE;
now
assume ('not' a 'or' 'not' b).z<>TRUE;
then ('not' a 'or' 'not' b).z=FALSE by XBOOLEAN:def 3;
then
A3: ('not' a).z 'or' ('not' b).z=FALSE by BVFUNC_1:def 4;
('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3;
then (('not' a).z 'or' c.z) '&' (('not' b).z 'or' ('not' c).z) =c.z '&'
'not' c.z by A3,MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem
(a 'or' b) '&' ('not' a 'or' c) '<' b 'or' c
proof
let z be Element of Y;
A1: ((a 'or' b) '&' ('not' a 'or' c)).z =(a 'or' b).z '&' ('not' a 'or' c).z
by MARGREL1:def 20
.=(a.z 'or' b.z) '&' ('not' a 'or' c).z by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (('not' a).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'or' b) '&' ('not' a 'or' c)).z=TRUE;
now
assume (b 'or' c).z<>TRUE;
then (b 'or' c).z=FALSE by XBOOLEAN:def 3;
then
A3: b.z 'or' c.z=FALSE by BVFUNC_1:def 4;
c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
then (a.z 'or' b.z) '&' (('not' a).z 'or' c.z) =a.z '&' 'not' a.z by A3,
MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence contradiction by A2,A1;
end;
hence thesis;
end;
theorem Th918:
(a 'imp' b) '&' (c 'imp' d) '<' (a '&' c) 'imp' (b '&' d)
proof
let z be Element of Y;
A1: ((a 'imp' b) '&' (c 'imp' d)).z =(a 'imp' b).z '&' (c 'imp' d).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (c 'imp' d).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' c 'or' d).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' c 'or' d).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' c).z 'or' (d).z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (c 'imp' d)).z=TRUE;
now
A3: ('not' c).z=TRUE or ('not' c).z=FALSE by XBOOLEAN:def 3;
A4: (('not' a).z 'or' ('not' c).z)=TRUE or (('not' a).z 'or' ('not' c).z)=
FALSE by XBOOLEAN:def 3;
A5: (b.z '&' (d).z)=TRUE or (b.z '&' (d).z)=FALSE by XBOOLEAN:def 3;
A6: ((a '&' c) 'imp' (b '&' d)).z =('not'( a '&' c) 'or' (b '&' d)).z by
BVFUNC_4:8
.=('not'( a '&' c)).z 'or' (b '&' d).z by BVFUNC_1:def 4
.=('not' a 'or' 'not' c).z 'or' (b '&' d).z by BVFUNC_1:14
.=(('not' a).z 'or' ('not' c).z) 'or' (b '&' d).z by BVFUNC_1:def 4
.=(('not' a).z 'or' ('not' c).z) 'or' (b.z '&' (d).z) by MARGREL1:def 20;
assume
A7: ((a '&' c) 'imp' (b '&' d)).z<>TRUE;
now
per cases by A7,A6,A5,MARGREL1:12;
case
b.z=FALSE;
thus thesis by A2,A1,A6,A4,A3;
end;
case
d.z=FALSE;
thus thesis by A2,A1,A6,A4,A3;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
(a 'imp' b) '&' (a 'imp' c) '<' a 'imp' (b '&' c)
proof
(a 'imp' b) '&' (a 'imp' c) '<' (a '&' a) 'imp' (b '&' c) by Th918;
hence thesis;
end;
theorem Th20:
((a 'imp' c) '&' (b 'imp' c)) '<' (a 'or' b) 'imp' c
proof
((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c) = I_el Y by
Th9;
hence thesis by BVFUNC_1:16;
end;
theorem Th21:
(a 'imp' b) '&' (c 'imp' d) '<' (a 'or' c) 'imp' (b 'or' d)
proof
let z be Element of Y;
A1: ((a 'imp' b) '&' (c 'imp' d)).z =(('not' a 'or' b) '&' (c 'imp' d)).z by
BVFUNC_4:8
.=(('not' a 'or' b) '&' ('not' c 'or' d)).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' c 'or' d).z by MARGREL1:def 20
.=(('not' a).z 'or' b.z) '&' ('not' c 'or' d).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' c).z 'or' (d).z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (c 'imp' d)).z=TRUE;
now
A3: (d).z=TRUE or (d).z=FALSE by XBOOLEAN:def 3;
A4: (b.z 'or' (d).z)=TRUE or (b.z 'or' (d).z)=FALSE by XBOOLEAN:def 3;
A5: (('not' a).z '&' ('not' c).z)=TRUE or (('not' a).z '&' ('not' c).z)=
FALSE by XBOOLEAN:def 3;
A6: ((a 'or' c) 'imp' (b 'or' d)).z =('not'( a 'or' c) 'or' (b 'or' d)).z
by BVFUNC_4:8
.=(('not' a '&' 'not' c) 'or' (b 'or' d)).z by BVFUNC_1:13
.=('not' a '&' 'not' c).z 'or' (b 'or' d).z by BVFUNC_1:def 4
.=(('not' a).z '&' ('not' c).z) 'or' (b 'or' d).z by MARGREL1:def 20
.=(('not' a).z '&' ('not' c).z) 'or' (b.z 'or' (d).z) by BVFUNC_1:def 4;
assume
A7: ((a 'or' c) 'imp' (b 'or' d)).z<>TRUE;
now
per cases by A7,A6,A5,MARGREL1:12;
case
('not' a).z=FALSE;
thus thesis by A2,A1,A6,A4,A3;
end;
case
('not' c).z=FALSE;
thus thesis by A2,A1,A6,A4,A3;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
(a 'imp' b) '&' (a 'imp' c) '<' a 'imp' (b 'or' c)
proof
(a 'imp' b) '&' (a 'imp' c) '<' (a 'or' a) 'imp' (b 'or' c) by Th21;
hence thesis;
end;
theorem Th23:
for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds (
b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2)
'&' 'not'( a2 '&' c2) '<' (a2 'imp' a1)
proof
let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2 'or' c2) 'imp' 'not' a2) '&'
((b1 'or' c1) 'imp' 'not' a2) 'imp' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by
Th38;
A2: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (c1 'imp' c2) = I_el(Y)
by Lm4,BVFUNC_1:16;
A3: ((b1 'imp' b2) '&' (c1 'imp' c2)) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2
)) = I_el(Y) by Th21,BVFUNC_1:16;
(b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (b1 'imp' b2) = I_el(Y)
by Lm4,BVFUNC_1:16;
then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (b1 'imp' b2) '&' (c1 'imp' c2) = I_el(Y)
by A2,th18;
then
A4: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) = I_el(Y)
by A3,BVFUNC_5:9;
A5: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' 'not'( a2 '&' c2) = I_el(Y) by Lm1,
BVFUNC_1:16;
(b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' 'not'( a2 '&' b2) = I_el(Y) by Lm2,
BVFUNC_1:16;
then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ('not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) =
I_el(Y) by A5,th18;
then
A6: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' (
'not'( a2 '&' b2) '&' 'not' (a2 '&' c2)) = I_el(Y) by A4,th18;
'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) =('not' a2 'or' 'not' b2) '&'
'not'( a2 '&' c2) by BVFUNC_1:14
.=('not' b2 'or' 'not' a2) '&' ('not' c2 'or' 'not' a2) by BVFUNC_1:14
.=(b2 'imp' 'not' a2) '&' ('not' c2 'or' 'not' a2) by BVFUNC_4:8
.=(b2 'imp' 'not' a2) '&' (c2 'imp' 'not' a2) by BVFUNC_4:8
.=(b2 'or' c2) 'imp' 'not' a2 by Th75;
then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2
'or' c2) 'imp' 'not' a2) '&' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by A6,
Th12;
then
A7: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by
A1,BVFUNC_5:9;
(a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) =(a1 'or' (b1
'or' c1) ) '&' ((b1 'or' c1) 'imp' 'not' a2) & (a1 'or' (b1 'or' c1)) '&' ((b1
'or' c1) 'imp' 'not' a2) '<' (a1 'or' 'not' a2) by Th1,BVFUNC_1:8;
then
A8: (a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) 'imp' (a1 'or'
'not' a2) =I_el(Y) by BVFUNC_1:16;
(b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' b1 'or' c1) = I_el(Y) by Lm3,
BVFUNC_1:16;
then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' b1 'or' c1) '&' ((b1 'or' c1)
'imp' 'not' a2) = I_el(Y) by A7,th18;
then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' 'not' a2) = I_el(Y) by A8,
BVFUNC_5:9;
then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (a2 'imp' a1) = I_el(Y) by BVFUNC_4:8;
hence thesis by BVFUNC_1:16;
end;
Lm7: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ((a1 'imp'
a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (
b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&'
b2)) =I_el(Y)
proof
let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp'
'not'( a2 '&' c2) = I_el(Y) by Lm2,BVFUNC_1:16;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) = I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&'
( a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2
'&' c2)) 'imp' (b1 'imp' b2) = I_el(Y)
by Lm6,BVFUNC_1:16;
then
A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) '&' (b1 'imp' b2) = I_el(Y) by th18;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'or' b1 'or' c1) = I_el(Y) by Lm4,BVFUNC_1:16;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) = I_el(Y) by A2,
th18;
then
A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' c2) =
I_el(Y) by A1,th18;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp'
'not'( b2 '&' c2) = I_el(Y) by Lm1,BVFUNC_1:16;
hence thesis by A3,th18;
end;
Lm8: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ((a1 'imp'
a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (
c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( b2 '&' a2) '&' 'not'( b2 '&'
c2)) = I_el(Y)
proof
let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp'
'not'( a2 '&' b2) = I_el(Y) by Lm3,BVFUNC_1:16;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) = I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&'
( a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2
'&' c2)) 'imp' (c1 'imp' c2) = I_el(Y)
by Lm5,Lm6,BVFUNC_1:16;
then
A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) '&' (c1 'imp' c2) = I_el(Y) by th18;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'or' b1 'or' c1) = I_el(Y) by Lm4,BVFUNC_1:16;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) = I_el(Y) by A2,
th18;
then
A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) =
I_el(Y) by A1,th18;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp'
'not'( b2 '&' c2) = I_el(Y) by Lm1,BVFUNC_1:16;
hence thesis by A3,th18;
end;
Lm9: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ((a1 'imp'
a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((b1 'imp' b2) '&' (
c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&'
c2)) =I_el(Y)
proof
let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp'
'not'( a2 '&' b2) = I_el(Y) by Lm3,BVFUNC_1:16;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
b1 'imp' b2) = I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&'
( a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2
'&' c2)) 'imp' (c1 'imp' c2) = I_el(Y)
by Lm5,Lm6,BVFUNC_1:16;
then
A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
b1 'imp' b2) '&' (c1 'imp' c2) = I_el(Y) by th18;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a1 'or' b1 'or' c1) = I_el(Y) by Lm4,BVFUNC_1:16;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) = I_el(Y) by A2,
th18;
then
A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) =
I_el(Y) by A1,th18;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp'
'not'( a2 '&' c2) = I_el(Y) by Lm2,BVFUNC_1:16;
hence thesis by A3,th18;
end;
theorem
for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds (a1
'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&'
'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) '<' (a2 'imp' a1)
'&' (b2 'imp' b1) '&' (c2 'imp' c1)
proof
let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((
b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2)
'&' 'not'( a2 '&' c2)) =I_el(Y) by Lm9;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((
a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2)
'&' 'not'( c2 '&' b2)) =I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1
'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2)
'&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1
'or' c1) '&' 'not'( b2 '&' a2) '&' 'not'( b2 '&' c2)) =I_el(Y) by Lm7,Lm8;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((
a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2)
'&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1
'or' c1) '&' 'not'( b2 '&' a2) '&' 'not'( b2 '&' c2)) =I_el(Y) by th18;
then
A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((
a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2)
'&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1
'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) '&' ((b1 'imp' b2) '&' (
c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&'
c2)) =I_el(Y) by A1,th18;
A3: (a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( b2 '&' c2) 'imp' (b2 'imp' b1) = I_el(Y)
by Th23,BVFUNC_1:16;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2
'&' a2) '&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1
'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) '&' ((b1 'imp' b2
) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'(
a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&'
'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) =I_el(Y)
by Lm2,BVFUNC_1:16;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((
a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2)
'&' 'not'( b2 '&' c2)) =I_el(Y) by A2,BVFUNC_5:9;
then
A4: (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) 'imp' (b2
'imp' b1) = I_el(Y) by A3,BVFUNC_5:9;
(b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' (a2 'imp' a1) = I_el(Y)
by Th23,BVFUNC_1:16;
then (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not' (b2 '&' c2) 'imp' (a2
'imp' a1) = I_el(Y) by A1,BVFUNC_5:9;
then
A5: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a2 'imp' a1) '&' (b2 'imp' b1) = I_el(Y) by A4,th18;
(a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' c1 'or' b1) '&' 'not'( c2
'&' a2) '&' 'not'( c2 '&' b2) '<' (c2 'imp' c1) by Th23;
then (a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2
'&' a2) '&' 'not'( c2 '&' b2) '<' (c2 'imp' c1) by BVFUNC_1:8;
then
A6: (a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2
'&' a2) '&' 'not'( c2 '&' b2) 'imp' (c2 'imp' c1) = I_el(Y) by BVFUNC_1:16;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2
'&' a2) '&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1
'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) '&' ((b1 'imp' b2
) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'(
a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&'
'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) =I_el(Y) by Lm2,BVFUNC_1:16;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((
a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2)
'&' 'not'( c2 '&' b2)) =I_el(Y) by A2,BVFUNC_5:9;
then (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) 'imp' (c2
'imp' c1) = I_el(Y) by A6,BVFUNC_5:9;
then
((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (
a2 'imp' a1) '&' (b2 'imp' b1) '&' (c2 'imp' c1) = I_el(Y) by A5,th18;
hence thesis by BVFUNC_1:16;
end;
theorem Th25:
for a1,b1,a2,b2 being Function of Y,BOOLEAN holds ((a1
'imp' a2) '&' (b1 'imp' b2) '&' 'not'(a2 '&' b2)) 'imp' 'not' (a1 '&' b1)=I_el(
Y)
proof
let a1,b1,a2,b2 be Function of Y,BOOLEAN;
for z being Element of Y st ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'(
a2 '&' b2)).z=TRUE holds ('not'( a1 '&' b1)).z=TRUE
proof
let z be Element of Y;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z =((a1 'imp'
a2) '&' (b1 'imp' b2)).z '&' ('not'( a2 '&' b2)).z by MARGREL1:def 20
.=((a1 'imp' a2).z '&' (b1 'imp' b2).z) '&' ('not'( a2 '&' b2)).z by
MARGREL1:def 20
.=(('not' a1 'or' a2).z '&' (b1 'imp' b2).z) '&' ('not'( a2 '&' b2)).z
by BVFUNC_4:8
.=(('not' a1 'or' a2).z '&' ('not' b1 'or' b2).z) '&' ('not' (a2 '&'
b2)).z by BVFUNC_4:8
.=((('not' a1).z 'or' (a2).z) '&' ('not' b1 'or' b2).z) '&' ('not'( a2
'&' b2)).z by BVFUNC_1:def 4
.=((('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z)) '&' (
'not'( a2 '&' b2)).z by BVFUNC_1:def 4
.=((('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z)) '&' ((
'not' a2 'or' 'not' b2).z) by BVFUNC_1:14
.=((('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z)) '&' ((
'not' a2).z 'or' ('not' b2).z) by BVFUNC_1:def 4;
assume
A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z=TRUE;
now
A3: ('not' b1).z=TRUE or ('not' b1).z=FALSE by XBOOLEAN:def 3;
A4: ('not' a1).z=TRUE or ('not' a1).z=FALSE by XBOOLEAN:def 3;
A5: ('not'( a1 '&' b1)).z =('not' a1 'or' 'not' b1).z by BVFUNC_1:14
.=('not' a1).z 'or' ('not' b1).z by BVFUNC_1:def 4;
assume ('not'( a1 '&' b1)).z<>TRUE;
then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z =((b2).z
'&' (a2).z) '&' ('not' a2).z 'or' ((a2).z '&' (b2).z) '&' ('not' b2).z by A1,A5
,A4,A3,XBOOLEAN:8
.=b2.z '&' ((a2).z '&' ('not' a2).z) 'or' (a2).z '&' ((b2).z '&' (
'not' b2).z)
.=b2.z '&' ((a2).z '&' 'not' (a2).z) 'or' (a2).z '&' ((b2).z '&' (
'not' b2).z) by MARGREL1:def 19
.=b2.z '&' ((a2).z '&' 'not' (a2).z) 'or' (a2).z '&' ((b2).z '&'
'not' (b2).z) by MARGREL1:def 19
.=b2.z '&' FALSE 'or' (a2).z '&' ((b2).z '&' 'not' (b2).z) by
XBOOLEAN:138
.=FALSE 'or' FALSE '&' a2.z by XBOOLEAN:138
.=FALSE;
hence contradiction by A2;
end;
hence thesis;
end;
hence thesis by BVFUNC_1:16,def 12;
end;
theorem
for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds (a1
'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' b2) '&' 'not'(
a2 '&' c2) '&' 'not'( b2 '&' c2) '<' 'not'( a1 '&' b1) '&' 'not'( a1 '&' c1)
'&' 'not'( b1 '&' c1)
proof
let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2
'&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) =I_el(Y)
by Th38;
A2: ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) 'imp' 'not'( a1
'&' c1)=I_el(Y) by Th25;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp'
c2) '&' 'not'( a2 '&' c2)) =I_el(Y) by Th38;
then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2
'&' c2)) =I_el(Y) by A1,BVFUNC_5:9;
then
A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a1 '&' c1)=I_el(Y) by A2,BVFUNC_5:9;
A4: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) 'imp' 'not'( a1
'&' b1)=I_el(Y) by Th25;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2
'&' c2)) =I_el(Y) & ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2))
'imp' 'not'( b1 '&' c1)=I_el(Y) by Th25,Th38;
then
A5: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( b1 '&' c1)=I_el(Y) by BVFUNC_5:9;
((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp'
b2) '&' 'not'( a2 '&' b2)) =I_el(Y) by Th38;
then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2
'&' b2)) =I_el(Y) by A1,BVFUNC_5:9;
then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'(a1 '&' b1)=I_el(Y) by A4,BVFUNC_5:9;
then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'(a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a1 '&' b1) '&' 'not'( a1 '&' c1) =I_el(
Y) by A3,th18;
then
A6: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp'
a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp'
c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a1 '&' b1) '&' 'not'( a1 '&' c1) '&'
'not'( b1 '&' c1) =I_el(Y) by A5,th18;
(a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' b2)
'&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) =(a1 'imp' a2) '&' (a1 'imp' a2)
'&' ((b1 'imp' b2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2) '&' 'not'( a2 '&' b2)
'&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)
.=(((a1 'imp' a2) '&' (a1 'imp' a2) '&' (b1 'imp' b2)) '&' (b1 'imp' b2)
'&' ((c1 'imp' c2) '&' (c1 'imp' c2))) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&'
c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2)) '&' (b1 'imp'
b2) '&' ((c1 'imp' c2) '&' (c1 'imp' c2))) '&' 'not'( a2 '&' b2) '&' 'not'( a2
'&' c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=(((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2) '&' (b1 'imp' b2)
'&' (c1 'imp' c2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2
) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=(((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2) '&' ((b1 'imp' b2
) '&' (c1 'imp' c2)) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&'
c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=(((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2
)) '&' (a1 'imp' a2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&'
c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2)
) '&' ((a1 'imp' a2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&'
c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2)
) '&' ((a1 'imp' a2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' c2) '&' 'not'( a2 '&'
b2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2)
) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' 'not'( a2 '&'
b2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2)
) '&' 'not'( a2 '&' b2) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not' (a2 '&'
c2)) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' 'not'( a2 '&' b2) '&' ((b1 'imp'
b2) '&' (c1 'imp' c2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&'
c2)) '&' 'not'( b2 '&' c2) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((b1 'imp'
b2) '&' (c1 'imp' c2)) '&' 'not'( b2 '&' c2) '&' ((a1 'imp' a2) '&' (c1 'imp'
c2) '&' 'not' (a2 '&' c2)) by BVFUNC_1:4
.=((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((b1 'imp'
b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) '&' ((a1 'imp' a2) '&' (c1 'imp'
c2) '&' 'not'( a2 '&' c2)) by BVFUNC_1:4
.=(((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1
'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1
'imp' c2) '&' 'not'( b2 '&' c2))) by BVFUNC_1:4;
hence thesis by A6,BVFUNC_1:16;
end;
theorem
a '&' b '<' a by Lm1;
theorem
a '&' b '&' c '<' a & a '&' b '&' c '<' b by Lm2;
theorem
a '&' b '&' c '&' d '<' a & a '&' b '&' c '&' d '<' b by Lm3;
theorem
a '&' b '&' c '&' d '&' e '<' a & a '&' b '&' c '&' d '&' e '<' b by Lm4;
theorem
a '&' b '&' c '&' d '&' e '&' f '<' a & a '&' b '&' c '&' d '&' e '&'
f '<' b by Lm5;
theorem
a '&' b '&' c '&' d '&' e '&' f '&' g '<' a & a '&' b '&' c '&' d '&'
e '&' f '&' g '<' b by Lm6;
theorem Th33:
a '<' b & c '<' d implies a '&' c '<' b '&' d
proof
assume a '<' b & c '<' d;
then a 'imp' b = I_el Y & c 'imp' d = I_el Y by BVFUNC_1:16;
then (a '&' c) 'imp' (b '&' d) = I_el Y by tt;
hence thesis by BVFUNC_1:16;
end;
theorem
a '&' b '<' c implies a '&' 'not' c '<' 'not' b
proof
assume a '&' b '<' c;
then I_el Y = a '&' b 'imp' c by BVFUNC_1:16
.= 'not' (a '&' b) 'or' c by BVFUNC_4:8
.= 'not' a 'or' 'not' b 'or' c by BVFUNC_1:14
.= 'not' a 'or' 'not' 'not' c 'or' 'not' b by BVFUNC_1:8
.= 'not' (a '&' 'not' c) 'or' 'not' b by BVFUNC_1:14
.= a '&' 'not' c 'imp' 'not' b by BVFUNC_4:8;
hence thesis by BVFUNC_1:16;
end;
theorem
(a 'imp' c) '&' (b 'imp' c) '&' (a 'or' b) '<' c
proof
set K1 = (a 'imp' c) '&' (b 'imp' c);
K1 '<' (a 'or' b) 'imp' c by Th20;
then
A1: K1 '&' (a 'or' b) '<' ((a 'or' b) 'imp' c) '&' (a 'or' b) by Th33;
((a 'or' b) 'imp' c) '&' (a 'or' b) '<' c by Th2;
hence thesis by A1,BVFUNC_1:15;
end;
theorem
((a 'imp' c) 'or' (b 'imp' c)) '&' (a '&' b) '<' c
proof
(a 'imp' c) 'or' (b 'imp' c) = (a '&' b) 'imp' c by Th76;
hence thesis by Th2;
end;
theorem
a '<' b & c '<' d implies a 'or' c '<' b 'or' d
proof
assume a '<' b & c '<' d;
then a 'imp' b = I_el(Y) & c 'imp' d = I_el(Y) by BVFUNC_1:16;
then (a 'or' c) 'imp' (b 'or' d) = I_el(Y) by Th22;
hence thesis by BVFUNC_1:16;
end;
theorem Th38a:
a '<' a 'or' b
proof
a 'imp' (a 'or' b) = I_el Y by Th26;
hence thesis by BVFUNC_1:16;
end;
theorem
a '&' b '<' a 'or' b
proof
a '&' b '<' a & a '<' a 'or' b by Lm1,Th38a;
hence thesis by BVFUNC_1:15;
end;
begin :: BVFUNC10
reserve Y for non empty set;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'or' (b
'&' c) 'or' (c '&' a)= (a 'or' b) '&' (b 'or' c) '&' (c 'or' a)
proof
let a,b,c be Function of Y,BOOLEAN;
for z being Element of Y st ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z=
TRUE holds ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z=TRUE
proof
let z be Element of Y;
A1: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z =((a '&' b) 'or' (b '&' c)
).z 'or' (c '&' a).z by BVFUNC_1:def 4
.=(a '&' b).z 'or' (b '&' c).z 'or' (c '&' a).z by BVFUNC_1:def 4
.=(a.z '&' b.z) 'or' (b '&' c).z 'or' (c '&' a).z by MARGREL1:def 20
.=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c '&' a).z by
MARGREL1:def 20
.=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c.z '&' a.z) by
MARGREL1:def 20;
assume
A2: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z=TRUE;
now
A3: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z =((a 'or' b) '&' (b
'or' c)).z '&' (c 'or' a).z by MARGREL1:def 20
.=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z by MARGREL1:def 20
.=(a.z 'or' b.z) '&' (b 'or' c).z '&' (c 'or' a).z by
BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c 'or' a).z by
BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z)
by BVFUNC_1:def 4;
assume
A4: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z<>TRUE;
now
per cases by A4,A3,MARGREL1:12,XBOOLEAN:def 3;
case
A5: (a.z 'or' b.z) '&' (b.z 'or' c.z)=FALSE;
now
per cases by A5,MARGREL1:12;
case
A6: (a.z 'or' b.z)=FALSE;
b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
hence thesis by A2,A1,A6;
end;
case
A7: (b.z 'or' c.z)=FALSE;
c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
hence thesis by A2,A1,A7;
end;
end;
hence thesis;
end;
case
A8: (c.z 'or' a.z)=FALSE;
a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
hence thesis by A2,A1,A8;
end;
end;
hence thesis;
end;
hence thesis;
end;
then
A9: (a '&' b) 'or' (b '&' c) 'or' (c '&' a) '<' (a 'or' b) '&' (b 'or' c)
'&' (c 'or' a);
for z being Element of Y st ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).
z=TRUE holds ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z=TRUE
proof
let z be Element of Y;
A10: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z =((a 'or' b) '&' (b 'or'
c)).z '&' (c 'or' a).z by MARGREL1:def 20
.=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z by MARGREL1:def 20
.=(a.z 'or' b.z) '&' (b 'or' c).z '&' (c 'or' a).z by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c 'or' a).z by
BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) by
BVFUNC_1:def 4;
assume
A11: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z=TRUE;
now
A12: (b.z '&' c.z)=TRUE or (b.z '&' c.z)=FALSE by XBOOLEAN:def 3;
A13: (c.z '&' a.z)=TRUE or (c.z '&' a.z)=FALSE by XBOOLEAN:def 3;
A14: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z =((a '&' b) 'or' (b '&'
c)).z 'or' (c '&' a).z by BVFUNC_1:def 4
.=(a '&' b).z 'or' (b '&' c).z 'or' (c '&' a).z by BVFUNC_1:def 4
.=(a.z '&' b.z) 'or' (b '&' c).z 'or' (c '&' a).z by
MARGREL1:def 20
.=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c '&' a).z by
MARGREL1:def 20
.=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c.z '&' a.z) by
MARGREL1:def 20;
assume
A15: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z<>TRUE;
now
per cases by A15,A14,A13,A12,MARGREL1:12,XBOOLEAN:def 3;
case
a.z=FALSE & b.z=FALSE;
hence thesis by A11,A10;
end;
case
b.z=FALSE & c.z=FALSE;
hence thesis by A11,A10;
end;
case
c.z=FALSE & a.z=FALSE;
hence thesis by A11,A10;
end;
end;
hence thesis;
end;
hence thesis;
end;
then (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '<' (a '&' b) 'or' (b '&' c)
'or' (c '&' a);
hence thesis by A9,BVFUNC_1:15;
end;
Lm1: for a,b,c being Function of Y,BOOLEAN holds (a '&' 'not' b) 'or' (b
'&' 'not' c) 'or' (c '&' 'not' a) '<' (b '&' 'not' a) 'or' (c '&' 'not' b) 'or'
(a '&' 'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a)).z =((a '&'
'not' b) 'or' (b '&' 'not' c)).z 'or' (c '&' 'not' a).z by BVFUNC_1:def 4
.=(a '&' 'not' b).z 'or' (b '&' 'not' c).z 'or' (c '&' 'not' a).z by
BVFUNC_1:def 4
.=(a.z '&' ('not' b).z) 'or' (b '&' 'not' c).z 'or' (c '&' 'not' a).z
by MARGREL1:def 20
.=(a.z '&' ('not' b).z) 'or' (b.z '&' ('not' c).z) 'or' (c '&' 'not'
a).z by MARGREL1:def 20
.=(a.z '&' ('not' b).z) 'or' (b.z '&' ('not' c).z) 'or' (c.z '&' (
'not' a).z) by MARGREL1:def 20
.=(a.z '&' 'not' b.z) 'or' (b.z '&' ('not' c).z) 'or' (c.z '&' (
'not' a).z) by MARGREL1:def 19
.=(a.z '&' 'not' b.z) 'or' (b.z '&' 'not' c.z) 'or' (c.z '&' (
'not' a).z) by MARGREL1:def 19
.=(a.z '&' 'not' b.z) 'or' (b.z '&' 'not' c.z) 'or' (c.z '&'
'not' a.z) by MARGREL1:def 19;
assume
A2: ((a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a)).z=TRUE;
now
A3: (a.z '&' ('not' c).z)=TRUE or (a.z '&' ('not' c).z)=FALSE by
XBOOLEAN:def 3;
assume
A4: ((b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c)).z<> TRUE;
A5: (c.z '&' ('not' b).z)=TRUE or (c.z '&' ('not' b).z)=FALSE by
XBOOLEAN:def 3;
A6: ((b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c)).z =((b '&'
'not' a) 'or' (c '&' 'not' b)).z 'or' (a '&' 'not' c).z by BVFUNC_1:def 4
.=(b '&' 'not' a).z 'or' (c '&' 'not' b).z 'or' (a '&' 'not' c).z by
BVFUNC_1:def 4
.=(b.z '&' ('not' a).z) 'or' (c '&' 'not' b).z 'or' (a '&' 'not' c).
z by MARGREL1:def 20
.=(b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z) 'or' (a '&'
'not' c).z by MARGREL1:def 20
.=(b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z) 'or' (a.z '&'
('not' c).z) by MARGREL1:def 20;
((b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z))=TRUE or ((b.z
'&' ('not' a).z) 'or' (c.z '&' ('not' b).z))=FALSE by XBOOLEAN:def 3;
then
A7: b.z=FALSE or ('not' a).z=FALSE by A4,A6,A5,MARGREL1:12;
now
per cases by A4,A6,A3,MARGREL1:12;
case
a.z=FALSE;
hence thesis by A2,A1,A6,A7,MARGREL1:def 19;
end;
case
('not' c).z=FALSE;
then
A8: 'not' c.z=FALSE by MARGREL1:def 19;
then 'not' b.z=FALSE by A4,A6,A5,MARGREL1:def 19;
hence thesis by A2,A1,A6,A5,A8,MARGREL1:def 19;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a '&' 'not' b) 'or'
(b '&' 'not' c) 'or' (c '&' 'not' a)= (b '&' 'not' a) 'or' (c '&' 'not' b) 'or'
(a '&' 'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
(a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a) '<' (b '&'
'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c) & (b '&' 'not' a) 'or' (c
'&' 'not' b) 'or' (a '&' 'not' c) '<' (a '&' 'not' b) 'or' (b '&' 'not' c) 'or'
(c '&' 'not' a) by Lm1;
hence thesis by BVFUNC_1:15;
end;
Lm2: for a,b,c being Function of Y,BOOLEAN holds (a 'or' 'not' b) '&' (b
'or' 'not' c) '&' (c 'or' 'not' a) '<' (b 'or' 'not' a) '&' (c 'or' 'not' b)
'&' (a 'or' 'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a)).z =((a 'or'
'not' b) '&' (b 'or' 'not' c)).z '&' (c 'or' 'not' a).z by MARGREL1:def 20
.=(a 'or' 'not' b).z '&' (b 'or' 'not' c).z '&' (c 'or' 'not' a).z by
MARGREL1:def 20
.=(a.z 'or' ('not' b).z) '&' (b 'or' 'not' c).z '&' (c 'or' 'not' a).z
by BVFUNC_1:def 4
.=(a.z 'or' ('not' b).z) '&' (b.z 'or' ('not' c).z) '&' (c 'or'
'not' a).z by BVFUNC_1:def 4
.=(a.z 'or' ('not' b).z) '&' (b.z 'or' ('not' c).z) '&' (c.z 'or'
('not' a).z) by BVFUNC_1:def 4
.=(a.z 'or' 'not' b.z) '&' (b.z 'or' ('not' c).z) '&' (c.z 'or'
('not' a).z) by MARGREL1:def 19
.=(a.z 'or' 'not' b.z) '&' (b.z 'or' 'not' c.z) '&' (c.z 'or'
('not' a).z) by MARGREL1:def 19
.=(a.z 'or' 'not' b.z) '&' (b.z 'or' 'not' c.z) '&' (c.z 'or'
'not' a.z) by MARGREL1:def 19;
assume
A2: ((a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a)).z=TRUE;
now
A3: ((b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c)).z =((b
'or' 'not' a) '&' (c 'or' 'not' b)).z '&' (a 'or' 'not' c).z by MARGREL1:def 20
.=(b 'or' 'not' a).z '&' (c 'or' 'not' b).z '&' (a 'or' 'not' c).z by
MARGREL1:def 20
.=(b.z 'or' ('not' a).z) '&' (c 'or' 'not' b).z '&' (a 'or' 'not' c)
.z by BVFUNC_1:def 4
.=(b.z 'or' ('not' a).z) '&' (c.z 'or' ('not' b).z) '&' (a 'or'
'not' c).z by BVFUNC_1:def 4
.=(b.z 'or' ('not' a).z) '&' (c.z 'or' ('not' b).z) '&' (a.z
'or' ('not' c).z) by BVFUNC_1:def 4;
assume
A4: ((b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c)).z<> TRUE;
now
per cases by A4,A3,MARGREL1:12,XBOOLEAN:def 3;
case
A5: ((b.z 'or' ('not' a).z) '&' (c.z 'or' ('not' b).z))=FALSE;
now
per cases by A5,MARGREL1:12;
case
A6: (b.z 'or' ('not' a).z)=FALSE;
A7: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3;
then 'not' a.z=FALSE by A6,MARGREL1:def 19;
hence thesis by A2,A1,A6,A7,XBOOLEAN:138;
end;
case
A8: (c.z 'or' ('not' b).z)=FALSE;
A9: ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3;
then 'not' b.z=FALSE by A8,MARGREL1:def 19;
hence thesis by A2,A1,A8,A9,XBOOLEAN:138;
end;
end;
hence thesis;
end;
case
A10: (a.z 'or' ('not' c).z)=FALSE;
A11: ('not' c).z=TRUE or ('not' c).z=FALSE by XBOOLEAN:def 3;
then 'not' c.z=FALSE by A10,MARGREL1:def 19;
hence thesis by A2,A1,A10,A11,XBOOLEAN:138;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'or' 'not' b) '&'
(b 'or' 'not' c) '&' (c 'or' 'not' a)= (b 'or' 'not' a) '&' (c 'or' 'not' b)
'&' (a 'or' 'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a) '<' (b 'or'
'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c) & (b 'or' 'not' a) '&' (c
'or' 'not' b) '&' (a 'or' 'not' c) '<' (a 'or' 'not' b) '&' (b 'or' 'not' c)
'&' (c 'or' 'not' a) by Lm2;
hence thesis by BVFUNC_1:15;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a)=I_el(Y)
& (c 'imp' b)=I_el(Y) implies c 'imp' (a 'or' b)=I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
assume
A1: (c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y);
c 'imp' (a 'or' b) =(c 'imp' a) 'or' (c 'imp' b) by Th73
.=I_el(Y) by A1;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c)=I_el(Y)
& (b 'imp' c)=I_el(Y) implies (a '&' b) 'imp' c = I_el(Y)
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y) implies (a '&' b) 'imp' (c '&'
c)=I_el(Y) by tt;
hence thesis;
end;
theorem
for a1,a2,b1,b2,c1,c2 being Function of Y,BOOLEAN holds (a1
'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '<' (a2
'or' b2 'or' c2)
proof
let a1,a2,b1,b2,c1,c2 be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or'
c1)).z =((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2)).z '&' (a1 'or' b1
'or' c1).z by MARGREL1:def 20
.=((a1 'imp' a2) '&' (b1 'imp' b2)).z '&' (c1 'imp' c2).z '&' (a1 'or'
b1 'or' c1).z by MARGREL1:def 20
.=(a1 'imp' a2).z '&' (b1 'imp' b2).z '&' (c1 'imp' c2).z '&' (a1 'or'
b1 'or' c1).z by MARGREL1:def 20
.=('not' a1 'or' a2).z '&' (b1 'imp' b2).z '&' (c1 'imp' c2).z '&' (a1
'or' b1 'or' c1).z by BVFUNC_4:8
.=('not' a1 'or' a2).z '&' ('not' b1 'or' b2).z '&' (c1 'imp' c2).z '&'
(a1 'or' b1 'or' c1).z by BVFUNC_4:8
.=('not' a1 'or' a2).z '&' ('not' b1 'or' b2).z '&' ('not' c1 'or' c2).z
'&' (a1 'or' b1 'or' c1).z by BVFUNC_4:8
.=(('not' a1).z 'or' (a2).z) '&' ('not' b1 'or' b2).z '&' ('not' c1 'or'
c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_1:def 4
.=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' ('not'
c1 'or' c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_1:def 4
.=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not'
c1).z 'or' (c2).z) '&' (a1 'or' b1 'or' c1).z by BVFUNC_1:def 4
.=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not'
c1).z 'or' (c2).z) '&' ((a1 'or' b1).z 'or' (c1).z) by BVFUNC_1:def 4
.=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not'
c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by BVFUNC_1:def 4
.=('not' (a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not'
c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by MARGREL1:def 19
.=('not' (a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' (b2).z) '&' (('not'
c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by MARGREL1:def 19
.=('not' (a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' (b2).z) '&' ('not' (
c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by MARGREL1:def 19;
assume
A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1
'or' c1)).z=TRUE;
now
A3: b2.z=TRUE or b2.z=FALSE by XBOOLEAN:def 3;
A4: c2.z=TRUE or c2.z=FALSE by XBOOLEAN:def 3;
A5: ('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1 'or' c1) =('not'
a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' ('not' a1 '&' 'not' b1 '&'
'not' c1) '&' c1 by BVFUNC_1:12
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' ('not' a1
'&' 'not' b1 '&' ('not' c1 '&' c1)) by BVFUNC_1:4
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' ('not' a1
'&' 'not' b1 '&' O_el(Y)) by BVFUNC_4:5
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' O_el(Y)
by BVFUNC_1:5
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) by BVFUNC_1:9
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' ('not' a1 '&' 'not'
b1 '&' 'not' c1) '&' b1 by BVFUNC_1:12
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' 'not' a1 '&' 'not'
c1 '&' 'not' b1 '&' b1 by BVFUNC_1:4
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' 'not' a1 '&' 'not'
c1 '&' ('not' b1 '&' b1) by BVFUNC_1:4
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' 'not' a1 '&' 'not'
c1 '&' O_el(Y) by BVFUNC_4:5
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' O_el(Y) by BVFUNC_1:5
.=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 by BVFUNC_1:9
.='not' b1 '&' 'not' c1 '&' 'not' a1 '&' a1 by BVFUNC_1:4
.='not' b1 '&' 'not' c1 '&' ('not' a1 '&' a1) by BVFUNC_1:4
.='not' b1 '&' 'not' c1 '&' O_el(Y) by BVFUNC_4:5
.=O_el(Y) by BVFUNC_1:5;
A6: ((a2).z 'or' (b2).z)=TRUE or ((a2).z 'or' (b2).z)=FALSE by XBOOLEAN:def 3;
A7: (a2 'or' b2 'or' c2).z =(a2 'or' b2).z 'or' (c2).z by BVFUNC_1:def 4
.=(a2).z 'or' b2.z 'or' c2.z by BVFUNC_1:def 4;
assume (a2 'or' b2 'or' c2).z<>TRUE;
then
('not' (a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' (b2).z) '&' ('not' (
c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) =(('not' a1).z '&'
'not' (b1).z '&' 'not' (c1).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by A7,A6,A4
,A3,MARGREL1:def 19
.=(('not' a1).z '&' ('not' b1).z '&' 'not' (c1).z) '&' ((a1).z 'or' (
b1).z 'or' (c1).z) by MARGREL1:def 19
.=(('not' a1).z '&' ('not' b1).z '&' ('not' c1).z) '&' ((a1).z 'or' (
b1).z 'or' (c1).z) by MARGREL1:def 19
.=(('not' a1).z '&' ('not' b1).z '&' ('not' c1).z) '&' ((a1 'or' b1).z
'or' (c1).z) by BVFUNC_1:def 4
.=(('not' a1).z '&' ('not' b1).z '&' ('not' c1).z) '&' ((a1 'or' b1
'or' c1).z) by BVFUNC_1:def 4
.=(('not' a1 '&' 'not' b1).z '&' ('not' c1).z) '&' ((a1 'or' b1 'or'
c1).z) by MARGREL1:def 20
.=(('not' a1 '&' 'not' b1 '&' 'not' c1).z) '&' ((a1 'or' b1 'or' c1).z
) by MARGREL1:def 20
.=(('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1 'or' c1)).z by
MARGREL1:def 20;
hence contradiction by A2,A1,A5,BVFUNC_1:def 10;
end;
hence thesis;
end;
Lm3: for a1,a2,b1,b2 being Function of Y,BOOLEAN holds (a1 'imp' b1) '&'
(a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2) '<' (b1 'imp' a1) '&' (b2
'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2)
proof
let a1,a2,b1,b2 be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2))
.z =(('not' a1 'or' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not' (b1 '&' b2
)).z by BVFUNC_4:8
.=(('not' a1 'or' b1) '&' ('not' a2 'or' b2) '&' (a1 'or' a2) '&' 'not'
(b1 '&' b2)).z by BVFUNC_4:8
.=(('not' a1 'or' b1) '&' ('not' a2 'or' b2) '&' (a1 'or' a2)).z '&' (
'not'( b1 '&' b2)).z by MARGREL1:def 20
.=(('not' a1 'or' b1) '&' ('not' a2 'or' b2)).z '&' (a1 'or' a2).z '&' (
'not'( b1 '&' b2)).z by MARGREL1:def 20
.=(('not' a1 'or' b1) '&' ('not' a2 'or' b2)).z '&' (a1 'or' a2).z '&' (
'not' b1 'or' 'not' b2).z by BVFUNC_1:14
.=('not' a1 'or' b1).z '&' ('not' a2 'or' b2).z '&' (a1 'or' a2).z '&' (
'not' b1 'or' 'not' b2).z by MARGREL1:def 20
.=(('not' a1).z 'or' (b1).z) '&' ('not' a2 'or' b2).z '&' (a1 'or' a2).z
'&' ('not' b1 'or' 'not' b2).z by BVFUNC_1:def 4
.=(('not' a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' (a1 'or'
a2).z '&' ('not' b1 'or' 'not' b2).z by BVFUNC_1:def 4
.=(('not' a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' ((a1).z
'or' (a2).z) '&' ('not' b1 'or' 'not' b2).z by BVFUNC_1:def 4
.=(('not' a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' ((a1).z
'or' (a2).z) '&' (('not' b1).z 'or' ('not' b2).z) by BVFUNC_1:def 4
.=('not' (a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' ((a1).z
'or' (a2).z) '&' (('not' b1).z 'or' ('not' b2).z) by MARGREL1:def 19
.=('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z
'or' (a2).z) '&' (('not' b1).z 'or' ('not' b2).z) by MARGREL1:def 19
.=('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z
'or' (a2).z) '&' ('not' (b1).z 'or' ('not' b2).z) by MARGREL1:def 19
.=('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z
'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) by MARGREL1:def 19;
assume
A2: ((a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&'
b2)).z=TRUE;
now
A3: ((b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2
)).z =(('not' b1 'or' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not' (a1 '&'
a2)).z by BVFUNC_4:8
.=(('not' b1 'or' a1) '&' ('not' b2 'or' a2) '&' (b1 'or' b2) '&'
'not' (a1 '&' a2)).z by BVFUNC_4:8
.=(('not' b1 'or' a1) '&' ('not' b2 'or' a2) '&' (b1 'or' b2)).z '&' (
'not'( a1 '&' a2)).z by MARGREL1:def 20
.=(('not' b1 'or' a1) '&' ('not' b2 'or' a2)).z '&' (b1 'or' b2).z '&'
('not'( a1 '&' a2)).z by MARGREL1:def 20
.=('not' b1 'or' a1).z '&' ('not' b2 'or' a2).z '&' (b1 'or' b2).z '&'
('not'( a1 '&' a2)).z by MARGREL1:def 20
.=('not' b1 'or' a1).z '&' ('not' b2 'or' a2).z '&' (b1 'or' b2).z '&'
(('not' a1 'or' 'not' a2)).z by BVFUNC_1:14
.=(('not' b1).z 'or' (a1).z) '&' ('not' b2 'or' a2).z '&' (b1 'or' b2)
.z '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:def 4
.=(('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' (b1
'or' b2).z '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:def 4
.=(('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1).
z 'or' (b2).z) '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:def 4
.=(('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1).
z 'or' (b2).z) '&' (('not' a1).z 'or' ('not' a2).z) by BVFUNC_1:def 4;
assume ((b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1
'&' a2)).z<>TRUE;
then
A4: (('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1).z
'or' (b2).z) '&' (('not' a1).z 'or' ('not' a2).z)=FALSE by A3,XBOOLEAN:def 3;
now
per cases by A4,MARGREL1:12;
case
A5: (('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&'
((b1).z 'or' (b2).z)=FALSE;
now
per cases by A5,MARGREL1:12;
case
A6: (('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) =FALSE;
now
per cases by A6,MARGREL1:12;
case
A7: (('not' b1).z 'or' (a1).z)=FALSE;
A8: (a1).z=TRUE or (a1).z=FALSE by XBOOLEAN:def 3;
then 'not' (b1).z=FALSE by A7,MARGREL1:def 19;
then
('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z)
'&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) =((a2).z '&'
'not' (a2).z 'or' (a2).z '&' (b2).z) '&' 'not' (b2).z by A7,A8,XBOOLEAN:8
.=(FALSE 'or' (a2).z '&' (b2).z) '&' 'not' (b2).z by
XBOOLEAN:138
.=((a2).z '&' (b2).z) '&' 'not' (b2).z
.=(a2).z '&' ((b2).z '&' 'not' (b2).z)
.=FALSE '&' (a2).z by XBOOLEAN:138
.=FALSE;
hence thesis by A2,A1;
end;
case
A9: (('not' b2).z 'or' (a2).z)=FALSE;
A10: (a2).z=TRUE or (a2).z=FALSE by XBOOLEAN:def 3;
then 'not' (b2).z=FALSE by A9,MARGREL1:def 19;
then
('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z)
'&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) =((a1).z '&'
'not' (a1).z 'or' (a1).z '&' (b1).z) '&' 'not' (b1).z by A9,A10,XBOOLEAN:8
.=(FALSE 'or' (a1).z '&' (b1).z) '&' 'not' (b1).z by
XBOOLEAN:138
.=((a1).z '&' (b1).z) '&' 'not' (b1).z
.=(a1).z '&' ((b1).z '&' 'not' (b1).z)
.=FALSE '&' (a1).z by XBOOLEAN:138
.=FALSE;
hence thesis by A2,A1;
end;
end;
hence thesis;
end;
case
A11: b1.z 'or' b2.z=FALSE;
reconsider a2z = a2.z as boolean object;
reconsider a1z = a1.z as boolean object;
(b1).z=TRUE or (b1).z=FALSE by XBOOLEAN:def 3;
then
('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&'
((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) ='not' (a1).z '&' (
'not' (a2).z '&' ((a1).z 'or' (a2).z)) by A11
.='not' (a1).z '&' ('not' (a2).z '&' (a1).z 'or' 'not' a2z '&'
a2z) by XBOOLEAN:8
.='not' (a1).z '&' ('not' (a2).z '&' (a1).z 'or' FALSE) by
XBOOLEAN:138
.='not' (a1).z '&' ((a1).z '&' 'not' (a2).z)
.='not' a1z '&' a1z '&' 'not' (a2).z
.=FALSE '&' 'not' (a2).z by XBOOLEAN:138
.=FALSE;
hence thesis by A2,A1;
end;
end;
hence thesis;
end;
case
A12: (('not' a1).z 'or' ('not' a2).z)=FALSE;
('not' a2).z=TRUE or ('not' a2).z=FALSE by XBOOLEAN:def 3;
then 'not' (a1).z=FALSE & 'not' (a2).z=FALSE by A12,MARGREL1:def 19;
then ('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((
a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) =(b1).z '&' ((b2).z '&'
('not' (b1).z 'or' 'not' (b2).z))
.=(b1).z '&' (b2.z '&' 'not' (b1).z 'or' (b2).z '&' 'not' (b2).z
) by XBOOLEAN:8
.=(b1).z '&' ((b2).z '&' 'not' (b1).z 'or' FALSE) by XBOOLEAN:138
.=(b1).z '&' ('not' (b1).z '&' (b2).z)
.=(b1).z '&' 'not' (b1).z '&' (b2).z
.=FALSE '&' (b2).z by XBOOLEAN:138
.=FALSE;
hence thesis by A2,A1;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
for a1,a2,b1,b2 being Function of Y,BOOLEAN holds (a1 'imp' b1)
'&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2)= (b1 'imp' a1) '&' (b2
'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2)
proof
let a1,a2,b1,b2 be Function of Y,BOOLEAN;
(a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2)
'<' ( b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2) & (
b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2) '<' (a1
'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2) by Lm3;
hence thesis by BVFUNC_1:15;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds (a 'or' b) '&' (c
'or' d) = (a '&' c) 'or' (a '&' d) 'or' (b '&' c) 'or' (b '&' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
(a 'or' b) '&' (c 'or' d) =((a 'or' b) '&' c) 'or' ((a 'or' b) '&' d) by
BVFUNC_1:12
.=(a '&' c) 'or' (b '&' c) 'or' ((a 'or' b) '&' d) by BVFUNC_1:12
.=(a '&' c) 'or' (b '&' c) 'or' ((a '&' d) 'or' (b '&' d)) by BVFUNC_1:12
.=(a '&' c) 'or' (b '&' c) 'or' (a '&' d) 'or' (b '&' d) by BVFUNC_1:8
.=(a '&' c) 'or' (a '&' d) 'or' (b '&' c) 'or' (b '&' d) by BVFUNC_1:8;
hence thesis;
end;
theorem
for a1,a2,b1,b2,b3 being Function of Y,BOOLEAN holds (a1 '&' a2)
'or' (b1 '&' b2 '&' b3)= (a1 'or' b1) '&' (a1 'or' b2) '&' (a1 'or' b3) '&' (a2
'or' b1) '&' (a2 'or' b2) '&' (a2 'or' b3)
proof
let a1,a2,b1,b2,b3 be Function of Y,BOOLEAN;
(a1 'or' b1) '&' (a1 'or' b2) '&' (a1 'or' b3) '&' (a2 'or' b1) '&' (a2
'or' b2) '&' (a2 'or' b3) =(a1 'or' (b1 '&' b2)) '&' (a1 'or' b3) '&' (a2 'or'
b1) '&' (a2 'or' b2) '&' (a2 'or' b3) by BVFUNC_1:11
.=(a1 'or' (b1 '&' b2 '&' b3)) '&' (a2 'or' b1) '&' (a2 'or' b2) '&' (a2
'or' b3) by BVFUNC_1:11
.=(a1 'or' (b1 '&' b2 '&' b3)) '&' ((a2 'or' b1) '&' (a2 'or' b2)) '&' (
a2 'or' b3) by BVFUNC_1:4
.=(a1 'or' (b1 '&' b2 '&' b3)) '&' (((a2 'or' b1) '&' (a2 'or' b2)) '&'
(a2 'or' b3)) by BVFUNC_1:4
.=(a1 'or' (b1 '&' b2 '&' b3)) '&' ((a2 'or' (b1 '&' b2)) '&' (a2 'or'
b3)) by BVFUNC_1:11
.=(a1 'or' (b1 '&' b2 '&' b3)) '&' (a2 'or' (b1 '&' b2 '&' b3)) by
BVFUNC_1:11
.=(a1 '&' a2) 'or' (b1 '&' b2 '&' b3) by BVFUNC_1:11;
hence thesis;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '&' (c 'imp' d)= (a 'imp' (b '&' c '&' d)) '&' (b 'imp' (c '&' d)) '&'
(c 'imp' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
A1: (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) =(a 'imp' b) '&' (b 'imp' c)
'&' (a 'imp' c) '&' (c 'imp' d) by Th12
.=(a 'imp' b) '&' (b 'imp' c) '&' ((a 'imp' c) '&' (c 'imp' d)) by
BVFUNC_1:4
.=(a 'imp' b) '&' (b 'imp' c) '&' ((a 'imp' c) '&' (c 'imp' d) '&' (a
'imp' d)) by Th12
.=(a 'imp' b) '&' (b 'imp' c) '&' ((a 'imp' c) '&' (c 'imp' d)) '&' (a
'imp' d) by BVFUNC_1:4
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a
'imp' d) by BVFUNC_1:4
.=(a 'imp' b) '&' ((b 'imp' c) '&' (c 'imp' d)) '&' (a 'imp' c) '&' (a
'imp' d) by BVFUNC_1:4
.=(a 'imp' b) '&' (((b 'imp' c) '&' (c 'imp' d)) '&' (b 'imp' d)) '&' (a
'imp' c) '&' (a 'imp' d) by Th12
.=(a 'imp' b) '&' ((b 'imp' c) '&' (c 'imp' d)) '&' (b 'imp' d) '&' (a
'imp' c) '&' (a 'imp' d) by BVFUNC_1:4
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (b 'imp' d) '&' (a
'imp' c) '&' (a 'imp' d) by BVFUNC_1:4
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (b
'imp' d) '&' (a 'imp' d) by BVFUNC_1:4
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a
'imp' d) '&' (b 'imp' d) by BVFUNC_1:4;
(a 'imp' (b '&' c '&' d)) '&' (b 'imp' (c '&' d)) '&' (c 'imp' d) =(
'not' a 'or' (b '&' c '&' d)) '&' (b 'imp' (c '&' d)) '&' (c 'imp' d) by
BVFUNC_4:8
.=('not' a 'or' (b '&' c '&' d)) '&' ('not' b 'or' (c '&' d)) '&' (c
'imp' d) by BVFUNC_4:8
.=('not' a 'or' (b '&' c '&' d)) '&' ('not' b 'or' (c '&' d)) '&' ('not'
c 'or' d) by BVFUNC_4:8
.=(('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' a 'or' d)) '&' (
'not' b 'or' (c '&' d)) '&' ('not' c 'or' d) by BVFUNC_5:39
.=(('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' a 'or' d)) '&' ((
'not' b 'or' c) '&' ('not' b 'or' d)) '&' ('not' c 'or' d) by BVFUNC_1:11
.=('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not'
b 'or' c) '&' ('not' b 'or' d) '&' ('not' c 'or' d) by BVFUNC_1:4
.=('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' b 'or' c) '&' ('not'
a 'or' d) '&' ('not' b 'or' d) '&' ('not' c 'or' d) by BVFUNC_1:4
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' a 'or' c) '&' ('not'
a 'or' d) '&' ('not' b 'or' d) '&' ('not' c 'or' d) by BVFUNC_1:4
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' a 'or' c) '&' ('not'
a 'or' d) '&' ('not' c 'or' d) '&' ('not' b 'or' d) by BVFUNC_1:4
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' a 'or' c) '&' ('not'
c 'or' d) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_1:4
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' d) '&' ('not'
a 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_1:4
.=(a 'imp' b) '&' ('not' b 'or' c) '&' ('not' c 'or' d) '&' ('not' a
'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8
.=(a 'imp' b) '&' (b 'imp' c) '&' ('not' c 'or' d) '&' ('not' a 'or' c)
'&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' ('not' a 'or' c) '&' (
'not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' ('not'
a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a
'imp' d) '&' ('not' b 'or' d) by BVFUNC_4:8
.=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a
'imp' d) '&' (b 'imp' d) by BVFUNC_4:8;
hence thesis by A1;
end;
theorem
for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' c) '&' (b
'imp' d) '&' (a 'or' b) '<' (c 'or' d)
proof
let a,b,c,d be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' c) '&' (b 'imp' d) '&' (a 'or' b)).z =((a 'imp' c) '&' (b
'imp' d)).z '&' (a 'or' b).z by MARGREL1:def 20
.=(a 'imp' c).z '&' (b 'imp' d).z '&' (a 'or' b).z by MARGREL1:def 20
.=('not' a 'or' c).z '&' (b 'imp' d).z '&' (a 'or' b).z by BVFUNC_4:8
.=('not' a 'or' c).z '&' ('not' b 'or' d).z '&' (a 'or' b).z by BVFUNC_4:8
.=(('not' a).z 'or' c.z) '&' ('not' b 'or' d).z '&' (a 'or' b).z by
BVFUNC_1:def 4
.=(('not' a).z 'or' c.z) '&' (('not' b).z 'or' (d).z) '&' (a 'or' b).z
by BVFUNC_1:def 4
.=(('not' a).z 'or' c.z) '&' (('not' b).z 'or' (d).z) '&' (a.z 'or'
b.z) by BVFUNC_1:def 4
.=('not' a.z 'or' c.z) '&' (('not' b).z 'or' (d).z) '&' (a.z 'or'
b.z) by MARGREL1:def 19
.=('not' a.z 'or' c.z) '&' ('not' b.z 'or' (d).z) '&' (a.z 'or'
b.z) by MARGREL1:def 19;
reconsider bz = b.z as boolean object;
reconsider az = a.z as boolean object;
assume
A2: ((a 'imp' c) '&' (b 'imp' d) '&' (a 'or' b)).z=TRUE;
now
assume (c 'or' d).z<>TRUE;
then (c 'or' d).z=FALSE by XBOOLEAN:def 3;
then
A3: c.z 'or' (d).z=FALSE by BVFUNC_1:def 4;
(d).z=TRUE or (d).z=FALSE by XBOOLEAN:def 3;
then
('not' a.z 'or' c.z) '&' ('not' b.z 'or' (d).z) '&' (a.z 'or'
b.z) ='not' a.z '&' ('not' b.z '&' (b.z 'or' a.z)) by A3
.='not' a.z '&' ( ('not' bz '&' bz 'or' 'not' b.z '&' a.z)) by
XBOOLEAN:8
.='not' a.z '&' ( (FALSE 'or' 'not' b.z '&' a.z)) by XBOOLEAN:138
.='not' a.z '&' (a.z '&' 'not' b.z)
.='not' az '&' az '&' 'not' b.z
.=FALSE '&' 'not' b.z by XBOOLEAN:138
.=FALSE;
hence thesis by A2,A1;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) 'imp'
'not' c) '&' a '&' c '<' 'not' b
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (((a '&' b) 'imp' 'not' c) '&' a '&' c).z =(((a '&' b) 'imp' 'not' c)
'&' a).z '&' c.z by MARGREL1:def 20
.=((a '&' b) 'imp' 'not' c).z '&' a.z '&' c.z by MARGREL1:def 20
.=('not'( a '&' b) 'or' 'not' c).z '&' a.z '&' c.z by BVFUNC_4:8
.=(('not' a 'or' 'not' b) 'or' 'not' c).z '&' a.z '&' c.z by
BVFUNC_1:14
.=(('not' a 'or' 'not' b).z 'or' ('not' c).z) '&' a.z '&' c.z by
BVFUNC_1:def 4
.=((('not' a).z 'or' ('not' b).z) 'or' ('not' c).z) '&' a.z '&' c.z
by BVFUNC_1:def 4;
reconsider cz = c.z as boolean object;
assume
A2: (((a '&' b) 'imp' 'not' c) '&' a '&' c).z=TRUE;
now
assume ('not' b).z<>TRUE;
then ('not' b).z=FALSE by XBOOLEAN:def 3;
then
((('not' a).z 'or' ('not' b).z) 'or' ('not' c).z) '&' a.z '&' c.z
=('not' a.z 'or' ('not' c).z) '&' a.z '&' c.z by MARGREL1:def 19
.=a.z '&' ('not' a.z 'or' 'not' c.z) '&' c.z by MARGREL1:def 19
.=(a.z '&' 'not' a.z 'or' a.z '&' 'not' c.z) '&' c.z by
XBOOLEAN:8
.=(FALSE 'or' a.z '&' 'not' c.z) '&' c.z by XBOOLEAN:138
.=(a.z '&' 'not' c.z) '&' c.z
.=a.z '&' ('not' cz '&' cz)
.=FALSE '&' a.z by XBOOLEAN:138
.=FALSE;
hence thesis by A2,A1;
end;
hence thesis;
end;
theorem
for a1,a2,a3,b1,b2,b3 being Function of Y,BOOLEAN holds (a1 '&'
a2 '&' a3) 'imp' (b1 'or' b2 'or' b3)= ('not' b1 '&' 'not' b2 '&' a3) 'imp' (
'not' a1 'or' 'not' a2 'or' b3)
proof
let a1,a2,a3,b1,b2,b3 be Function of Y,BOOLEAN;
('not' b1 '&' 'not' b2 '&' a3) 'imp' ('not' a1 'or' 'not' a2 'or' b3) =
'not'( 'not' b1 '&' 'not' b2 '&' a3) 'or' ('not' a1 'or' 'not' a2 'or' b3)
by BVFUNC_4:8
.=('not' 'not' b1 'or' 'not' 'not' b2 'or' 'not' a3) 'or' ('not' a1 'or'
'not' a2 'or' b3) by BVFUNC_5:37
.=b1 'or' b2 'or' 'not' a3 'or' ('not' a1 'or' 'not' a2) 'or' b3 by
BVFUNC_1:8
.=b1 'or' b2 'or' (('not' a1 'or' 'not' a2) 'or' 'not' a3) 'or' b3 by
BVFUNC_1:8
.=(('not' a1 'or' 'not' a2) 'or' 'not' a3) 'or' ((b1 'or' b2) 'or' b3)
by BVFUNC_1:8
.='not'( a1 '&' a2 '&' a3) 'or' (b1 'or' b2 'or' b3) by BVFUNC_5:37
.=(a1 '&' a2 '&' a3) 'imp' (b1 'or' b2 'or' b3) by BVFUNC_4:8;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '&' (c 'imp' a) = (a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not'
c)
proof
let a,b,c be Function of Y,BOOLEAN;
for z being Element of Y st ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)
).z=TRUE holds ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z=TRUE
proof
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z =((a 'imp' b) '&' (b
'imp' c)).z '&' (c 'imp' a).z by MARGREL1:def 20
.=(a 'imp' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' (c 'imp' a).z by
BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' ('not' c 'or' a).z by
BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z '&' ('not' c 'or' a)
.z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' ('not' c
'or' a).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not' c)
.z 'or' a.z) by BVFUNC_1:def 4
.=('not' a.z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not' c)
.z 'or' a.z) by MARGREL1:def 19
.=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' (('not' c)
.z 'or' a.z) by MARGREL1:def 19
.=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' ('not' (c)
.z 'or' a.z) by MARGREL1:def 19;
assume
A2: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z=TRUE;
now
A3: ('not' a.z '&' 'not' b.z '&' 'not' c.z)=TRUE or ('not' a.z
'&' 'not' b.z '&' 'not' c.z)=FALSE by XBOOLEAN:def 3;
assume
A4: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z<>TRUE;
A5: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z =(a '&' b
'&' c).z 'or' ('not' a '&' 'not' b '&' 'not' c).z by BVFUNC_1:def 4
.=((a '&' b).z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c).z
by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c)
.z by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' (('not' a '&' 'not' b).z '&' (
'not' c).z) by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' (('not' a).z '&' ('not' b).z '&'
('not' c).z) by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' ('not' b).z '&'
('not' c).z) by MARGREL1:def 19
.=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&'
('not' c).z) by MARGREL1:def 19
.=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&'
'not' c.z) by MARGREL1:def 19;
A6: (a.z '&' b.z '&' c.z)=TRUE or (a.z '&' b.z '&' c.z)=
FALSE by XBOOLEAN:def 3;
now
per cases by A4,A5,A6,MARGREL1:12;
case
A7: (a.z '&' b.z)=FALSE;
now
per cases by A7,MARGREL1:12;
case
A8: a.z=FALSE;
now
per cases by A4,A5,A3,A8,MARGREL1:12;
case
'not' b.z=FALSE;
hence thesis by A2,A1,A8,XBOOLEAN:138;
end;
case
'not' c.z=FALSE;
hence thesis by A2,A1,A8;
end;
end;
hence thesis;
end;
case
A9: b.z=FALSE;
now
per cases by A4,A5,A3,A9,MARGREL1:12;
case
'not' a.z=FALSE;
hence thesis by A2,A1,A9;
end;
case
'not' c.z=FALSE;
hence thesis by A2,A1,A9,XBOOLEAN:138;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
case
A10: c.z=FALSE;
now
per cases by A4,A5,A3,A10,MARGREL1:12;
case
'not' a.z=FALSE;
hence thesis by A2,A1,A10,XBOOLEAN:138;
end;
case
'not' b.z=FALSE;
hence thesis by A2,A1,A10;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
hence thesis;
end;
then
A11: (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '<' (a '&' b '&' c) 'or' (
'not' a '&' 'not' b '&' 'not' c);
for z being Element of Y st ((a '&' b '&' c) 'or' ('not' a '&' 'not' b
'&' 'not' c)).z=TRUE holds ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z=TRUE
proof
let z be Element of Y;
A12: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z =(a '&' b
'&' c).z 'or' ('not' a '&' 'not' b '&' 'not' c).z by BVFUNC_1:def 4
.=((a '&' b).z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c).z by
MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c).z
by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' (('not' a '&' 'not' b).z '&' ('not'
c).z) by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' (('not' a).z '&' ('not' b).z '&' (
'not' c).z) by MARGREL1:def 20
.=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' ('not' b).z '&' (
'not' c).z) by MARGREL1:def 19
.=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&' (
'not' c).z) by MARGREL1:def 19
.=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&'
'not' c.z) by MARGREL1:def 19;
assume
A13: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z=TRUE;
now
A14: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z =((a 'imp' b) '&' (
b 'imp' c)).z '&' (c 'imp' a).z by MARGREL1:def 20
.=(a 'imp' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' (c 'imp' a).z by
BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' ('not' c 'or' a).z
by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z '&' ('not' c 'or'
a).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' ('not' c
'or' a).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not'
c).z 'or' a.z) by BVFUNC_1:def 4
.=('not' a.z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not'
c).z 'or' a.z) by MARGREL1:def 19
.=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' (('not'
c).z 'or' a.z) by MARGREL1:def 19
.=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' ('not' (
c).z 'or' a.z) by MARGREL1:def 19;
assume ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z<>TRUE;
then
A15: ('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' ('not' (c
).z 'or' a.z)=FALSE by A14,XBOOLEAN:def 3;
now
per cases by A15,MARGREL1:12;
case
A16: ('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z)= FALSE;
now
per cases by A16,MARGREL1:12;
case
A17: ('not' a.z 'or' b.z)=FALSE;
b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
hence thesis by A13,A12,A17;
end;
case
A18: ('not' b.z 'or' c.z)=FALSE;
c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
hence thesis by A13,A12,A18;
end;
end;
hence thesis;
end;
case
A19: ('not' c.z 'or' a.z)=FALSE;
a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
hence thesis by A13,A12,A19;
end;
end;
hence thesis;
end;
hence thesis;
end;
then (a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c) '<' (a 'imp' b)
'&' (b 'imp' c) '&' (c 'imp' a);
hence thesis by A11,BVFUNC_1:15;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c)= (a '&' b '&' c)
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c) =(
'not' a 'or' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c) by
BVFUNC_4:8
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (c 'imp' a) '&' (a 'or' b
'or' c) by BVFUNC_4:8
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a) '&' (a 'or'
b 'or' c) by BVFUNC_4:8
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a
'or' b 'or' c)) by BVFUNC_1:4
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a
'or' b) 'or' ('not' c 'or' a) '&' c) by BVFUNC_1:12
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a
'or' b) 'or' ('not' c '&' c 'or' a '&' c)) by BVFUNC_1:12
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a
'or' b) 'or' (O_el(Y) 'or' a '&' c)) by BVFUNC_4:5
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a
'or' b) 'or' (a '&' c)) by BVFUNC_1:9
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' ('not' c '&' b) 'or'
(a '&' c)) by BVFUNC_1:11
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' (a '&' c) 'or' (
'not' c '&' b)) by BVFUNC_1:8
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a '&' I_el(Y)) 'or' (a '&'
c) 'or' ('not' c '&' b)) by BVFUNC_1:6
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a '&' (I_el(Y) 'or' c))
'or' ('not' c '&' b)) by BVFUNC_1:12
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a '&' I_el(Y)) 'or' ('not'
c '&' b)) by BVFUNC_1:10
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' ('not' c '&' b)) by
BVFUNC_1:6
.=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a 'or' 'not' c) '&' (a
'or' b)) by BVFUNC_1:11
.=(a 'or' b) '&' (('not' a 'or' b) '&' ('not' b 'or' c)) '&' (a 'or'
'not' c) by BVFUNC_1:4
.=(a 'or' b) '&' ('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' 'not'
c) by BVFUNC_1:4
.=((a '&' 'not' a) 'or' b) '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by
BVFUNC_1:11
.=(O_el(Y) 'or' b) '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by BVFUNC_4:5
.=b '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by BVFUNC_1:9
.=(b '&' 'not' b 'or' b '&' c) '&' (a 'or' 'not' c) by BVFUNC_1:12
.=(O_el(Y) 'or' b '&' c) '&' (a 'or' 'not' c) by BVFUNC_4:5
.=(b '&' c) '&' (a 'or' 'not' c) by BVFUNC_1:9
.=(b '&' c) '&' a 'or' (b '&' c) '&' 'not' c by BVFUNC_1:12
.=(b '&' c) '&' a 'or' b '&' (c '&' 'not' c) by BVFUNC_1:4
.=(b '&' c) '&' a 'or' b '&' O_el(Y) by BVFUNC_4:5
.=(b '&' c) '&' a 'or' O_el(Y) by BVFUNC_1:5
.=(b '&' c) '&' a by BVFUNC_1:9
.=(a '&' b '&' c) by BVFUNC_1:4;
hence thesis;
end;
Lm4: for a,b,c being Function of Y,BOOLEAN holds ('not' a '&' b '&' c)
'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) '<' (a 'or' b) '&' 'not'(
a '&' b '&' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c)).z =(('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c)).z 'or' (a '&' b
'&' 'not' c).z by BVFUNC_1:def 4
.=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a '&' b '&'
'not' c).z by BVFUNC_1:def 4
.=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' ((a '&' b).z
'&' ('not' c).z) by MARGREL1:def 20
.=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a.z '&' (
b).z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a '&' b '&' c).z 'or' ((a '&' 'not' b).z '&' c.z) 'or' (a.z
'&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a '&' b '&' c).z 'or' (a.z '&' ('not' b).z '&' c.z) 'or' ((
a).z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=(('not' a '&' b).z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.z)
'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=(('not' a).z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.
z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.
z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19
.=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c.
z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19
.=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c.
z) 'or' (a.z '&' b.z '&' 'not' c.z) by MARGREL1:def 19;
assume
A2: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c)).z=TRUE;
now
A3: ((a 'or' b) '&' 'not'( a '&' b '&' c)).z =((a 'or' b) '&' ('not' a
'or' 'not' b 'or' 'not' c)).z by BVFUNC_5:37
.=(a 'or' b).z '&' ('not' a 'or' 'not' b 'or' 'not' c).z by
MARGREL1:def 20
.=(a.z 'or' b.z) '&' ('not' a 'or' 'not' b 'or' 'not' c).z by
BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (('not' a 'or' 'not' b).z 'or' ('not' c).z)
by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (('not' a).z 'or' ('not' b).z 'or' ('not' c).
z) by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' ('not' a.z 'or' ('not' b).z 'or' ('not' c).
z) by MARGREL1:def 19
.=(a.z 'or' b.z) '&' ('not' a.z 'or' 'not' b.z 'or' ('not' c).
z) by MARGREL1:def 19
.=(a.z 'or' b.z) '&' ('not' a.z 'or' 'not' b.z 'or' 'not' c.
z) by MARGREL1:def 19;
assume ((a 'or' b) '&' 'not'( a '&' b '&' c)).z<>TRUE;
then
A4: (a.z 'or' b.z) '&' ('not' a.z 'or' 'not' b.z 'or' 'not' c.z)
=FALSE by A3,XBOOLEAN:def 3;
now
per cases by A4,MARGREL1:12;
case
A5: (a.z 'or' b.z)=FALSE;
b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
hence thesis by A2,A1,A5;
end;
case
A6: ('not' a.z 'or' 'not' b.z 'or' 'not' c.z)=FALSE;
A7: 'not' b.z=TRUE or 'not' b.z=FALSE by XBOOLEAN:def 3;
'not' a.z 'or' 'not' b.z=TRUE or 'not' a.z 'or' 'not' b.z
=FALSE by XBOOLEAN:def 3;
hence thesis by A2,A1,A6,A7;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'or' b) '&' (b
'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c)= ('not' a '&' b '&' c) 'or' (a
'&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c)
proof
let a,b,c be Function of Y,BOOLEAN;
A1: (a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' ((c 'or' a) '&'
'not'( a '&' b '&' c)) =(a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&'
(c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4
.=(a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c) '&'
'not'( a '&' b '&' c) by BVFUNC_1:4
.=(a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' ('not'( a '&' b '&' c)
'&' 'not'( a '&' b '&' c)) by BVFUNC_1:4
.=(a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c);
for z being Element of Y st ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)
'&' 'not' (a '&' b '&' c)).z=TRUE holds (('not' a '&' b '&' c) 'or' (a '&'
'not' b '&' c) 'or' (a '&' b '&' 'not' c)).z=TRUE
proof
let z be Element of Y;
A2: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c)).z
=((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z '&' ('not' (a '&' b '&' c)).z by
MARGREL1:def 20
.=((a 'or' b) '&' (b 'or' c)).z '&' (c 'or' a).z '&' ('not'( a '&' b
'&' c)).z by MARGREL1:def 20
.=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z '&' ('not'( a '&' b
'&' c)).z by MARGREL1:def 20
.=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z '&' (('not' a 'or'
'not' b 'or' 'not' c)).z by BVFUNC_5:37
.=(a.z 'or' b.z) '&' (b 'or' c).z '&' (c 'or' a).z '&' (('not' a
'or' 'not' b 'or' 'not' c)).z by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c 'or' a).z '&' ((
'not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&'
(('not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&'
(('not' a 'or' 'not' b).z 'or' ('not' c).z) by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&'
(('not' a).z 'or' ('not' b).z 'or' ('not' c).z) by BVFUNC_1:def 4
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&'
('not' a.z 'or' ('not' b).z 'or' ('not' c).z) by MARGREL1:def 19
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&'
('not' a.z 'or' 'not' b.z 'or' ('not' c).z) by MARGREL1:def 19
.=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&'
('not' a.z 'or' 'not' b.z 'or' 'not' c.z) by MARGREL1:def 19;
assume
A3: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not' (a '&' b '&' c
)).z=TRUE;
now
A4: (a.z '&' b.z '&' 'not' c.z)=TRUE or (a.z '&' b.z '&' 'not'
c.z)=FALSE by XBOOLEAN:def 3;
assume
A5: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b
'&' 'not' c)).z<>TRUE;
A6: (a.z '&' 'not' b.z '&' c.z)=TRUE or (a.z '&' 'not' b.z '&'
c.z)=FALSE by XBOOLEAN:def 3;
A7: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c)).z =(('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c)).z 'or' (a '&' b
'&' 'not' c).z by BVFUNC_1:def 4
.=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a '&' b
'&' 'not' c).z by BVFUNC_1:def 4
.=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' ((a '&'
b).z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a.z
'&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a '&' b '&' c).z 'or' ((a '&' 'not' b).z '&' c.z) 'or' ((
a).z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a '&' b '&' c).z 'or' (a.z '&' ('not' b).z '&' c.z)
'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=(('not' a '&' b).z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.
z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=(('not' a).z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&'
c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20
.=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&'
c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19
.=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&'
c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19
.=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&'
c.z) 'or' (a.z '&' b.z '&' 'not' c.z) by MARGREL1:def 19;
A8: ('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' (c
).z)=TRUE or ('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&'
c.z)=FALSE by XBOOLEAN:def 3;
now
per cases by A5,A7,A8,A6,MARGREL1:12;
case
A9: 'not' a.z '&' b.z=FALSE;
now
per cases by A9,MARGREL1:12;
case
A10: 'not' a.z=FALSE;
now
per cases by A5,A7,A6,A10,MARGREL1:12;
case
'not' b.z=FALSE;
hence thesis by A3,A2,A7,A10;
end;
case
c.z=FALSE;
hence thesis by A3,A2,A7,A10;
end;
end;
hence thesis;
end;
case
A11: b.z=FALSE;
now
per cases by A5,A7,A6,A11,MARGREL1:12;
case
a.z=FALSE;
hence thesis by A3,A2,A11;
end;
case
c.z=FALSE;
hence thesis by A3,A2,A11;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
case
A12: c.z=FALSE;
now
per cases by A5,A7,A4,A12,MARGREL1:12;
case
a.z=FALSE;
hence thesis by A3,A2,A12;
end;
case
b.z=FALSE;
hence thesis by A3,A2,A12;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
hence thesis;
end;
then
A13: (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c) '<'
('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c);
('not' c '&' a '&' b) 'or' (c '&' 'not' a '&' b) 'or' (c '&' a '&'
'not' b) '<' (c 'or' a) '&' 'not'( c '&' a '&' b) by Lm4;
then ('not' c '&' a '&' b) 'or' (c '&' 'not' a '&' b) 'or' (c '&' a '&'
'not' b) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then (a '&' b '&' 'not' c) 'or' (c '&' 'not' a '&' b) 'or' (c '&' a '&'
'not' b) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then (a '&' b '&' 'not' c) 'or' ('not' a '&' b '&' c) 'or' (c '&' a '&'
'not' b) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then (a '&' b '&' 'not' c) 'or' ('not' a '&' b '&' c) 'or' (a '&' 'not' b
'&' c) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:8;
then
A14: ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (c 'or' a) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:16;
('not' b '&' c '&' a) 'or' (b '&' 'not' c '&' a) 'or' (b '&' c '&'
'not' a) '<' (b 'or' c) '&' 'not'( b '&' c '&' a) by Lm4;
then ('not' b '&' c '&' a) 'or' (b '&' 'not' c '&' a) 'or' (b '&' c '&'
'not' a) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then ('not' b '&' c '&' a) 'or' (b '&' 'not' c '&' a) 'or' ('not' a '&' b
'&' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then (a '&' 'not' b '&' c) 'or' (b '&' 'not' c '&' a) 'or' ('not' a '&' b
'&' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'or' ('not' a '&' b
'&' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:8;
then
A15: ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (b 'or' c) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:16;
('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) '<' (a 'or' b) '&' 'not'( a '&' b '&' c) by Lm4;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (a 'or' b) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:16;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (a 'or' b) '&' 'not'( a '&' b '&' c) '&' ((b 'or' c) '&' 'not'(
a '&' b '&' c)) = I_el(Y) by A15,th18;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (a 'or' b) '&' 'not'( a '&' b '&' c) '&' (b 'or' c) '&' 'not'( a
'&' b '&' c) = I_el(Y) by BVFUNC_1:4;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' 'not'( a
'&' b '&' c) = I_el(Y) by BVFUNC_1:4;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (a 'or' b) '&' (b 'or' c) '&' ('not'( a '&' b '&' c) '&' 'not'(
a '&' b '&' c)) = I_el(Y) by BVFUNC_1:4;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) 'imp' (a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' ((c 'or'
a) '&' 'not'( a '&' b '&' c)) = I_el(Y) by A14,th18;
then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&'
'not' c) '<' (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c)
by A1,BVFUNC_1:16;
hence thesis by A13,BVFUNC_1:15;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' (a 'imp' (b '&' c))
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
now
A3: (a 'imp' (b '&' c)).z =('not' a 'or' (b '&' c)).z by BVFUNC_4:8
.=('not' a).z 'or' (b '&' c).z by BVFUNC_1:def 4
.=('not' a).z 'or' (b.z '&' c.z) by MARGREL1:def 20
.='not' a.z 'or' (b.z '&' c.z) by MARGREL1:def 19;
assume
A4: (a 'imp' (b '&' c)).z<>TRUE;
'not' a.z=TRUE or 'not' a.z=FALSE by XBOOLEAN:def 3;
then
A5: ('not' a).z=FALSE by A4,A3,MARGREL1:def 19;
A6: (b.z '&' c.z)=TRUE or (b.z '&' c.z)=FALSE by XBOOLEAN:def 3;
now
per cases by A4,A3,A6,MARGREL1:12;
case
b.z=FALSE;
hence thesis by A2,A1,A5;
end;
case
c.z=FALSE;
then (('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&'
'not' b.z by A5,MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence thesis by A2,A1;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' ((a 'or' b) 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
now
A3: c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
assume
A4: ((a 'or' b) 'imp' c).z<>TRUE;
A5: ((a 'or' b) 'imp' c).z =('not'( a 'or' b) 'or' c).z by BVFUNC_4:8
.=(('not' a '&' 'not' b) 'or' c).z by BVFUNC_1:13
.=('not' a '&' 'not' b).z 'or' c.z by BVFUNC_1:def 4
.=(('not' a).z '&' ('not' b).z) 'or' c.z by MARGREL1:def 20;
A6: (('not' a).z '&' ('not' b).z)=TRUE or (('not' a).z '&' ('not' b).z)=
FALSE by XBOOLEAN:def 3;
now
per cases by A4,A5,A6,MARGREL1:12;
case
('not' a).z=FALSE;
then (('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&'
'not' b.z by A4,A5,A3,MARGREL1:def 19
.=FALSE by XBOOLEAN:138;
hence thesis by A2,A1;
end;
case
('not' b).z=FALSE;
then
(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =(('not' a)
.z 'or' b.z) '&' FALSE by A4,A5,XBOOLEAN:def 3
.=FALSE;
hence thesis by A2,A1;
end;
end;
hence thesis;
end;
hence thesis;
end;
theorem Th19:
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)
'&' (b 'imp' c) '<' (a 'imp' (b 'or' c))
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
now
assume (a 'imp' (b 'or' c)).z<>TRUE;
A3: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3;
A4: (b.z 'or' c.z)=TRUE or (b.z 'or' c.z)=FALSE by XBOOLEAN:def 3;
A5: c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3;
(a 'imp' (b 'or' c)).z =('not' a 'or' (b 'or' c)).z by BVFUNC_4:8
.=('not' a).z 'or' (b 'or' c).z by BVFUNC_1:def 4
.=('not' a).z 'or' (b.z 'or' c.z) by BVFUNC_1:def 4;
hence thesis by A2,A1,A3,A4,A5;
end;
hence thesis;
end;
theorem Th20:
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)
'&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c))
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
now
assume (a 'imp' (b 'or' 'not' c)).z<>TRUE;
A3: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3;
A4: (b.z 'or' ('not' c).z)=TRUE or (b.z 'or' ('not' c).z)= FALSE by
XBOOLEAN:def 3;
A5: ('not' c).z=TRUE or ('not' c).z=FALSE by XBOOLEAN:def 3;
(a 'imp' (b 'or' 'not' c)).z =('not' a 'or' (b 'or' 'not' c)).z by
BVFUNC_4:8
.=('not' a).z 'or' (b 'or' 'not' c).z by BVFUNC_1:def 4
.=('not' a).z 'or' (b.z 'or' ('not' c).z) by BVFUNC_1:def 4;
hence thesis by A2,A1,A3,A4,A5;
end;
hence thesis;
end;
theorem Th21:
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)
'&' (b 'imp' c) '<' (b 'imp' (c 'or' a))
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
now
assume (b 'imp' (c 'or' a)).z<>TRUE;
A3: ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3;
A4: (c.z 'or' a.z)=TRUE or (c.z 'or' a.z)=FALSE by XBOOLEAN:def 3;
A5: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
(b 'imp' (c 'or' a)).z =('not' b 'or' (c 'or' a)).z by BVFUNC_4:8
.=('not' b).z 'or' (c 'or' a).z by BVFUNC_1:def 4
.=('not' b).z 'or' (c.z 'or' a.z) by BVFUNC_1:def 4;
hence thesis by A2,A1,A3,A4,A5;
end;
hence thesis;
end;
theorem Th22:
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)
'&' (b 'imp' c) '<' (b 'imp' (c 'or' 'not' a))
proof
let a,b,c be Function of Y,BOOLEAN;
let z be Element of Y;
A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by
MARGREL1:def 20
.=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8
.=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8
.=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4
.=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4;
assume
A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE;
now
assume (b 'imp' (c 'or' 'not' a)).z<>TRUE;
A3: ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3;
A4: (c.z 'or' ('not' a).z)=TRUE or (c.z 'or' ('not' a).z)= FALSE by
XBOOLEAN:def 3;
A5: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3;
(b 'imp' (c 'or' 'not' a)).z =('not' b 'or' (c 'or' 'not' a)).z by
BVFUNC_4:8
.=('not' b).z 'or' (c 'or' 'not' a).z by BVFUNC_1:def 4
.=('not' b).z 'or' (c.z 'or' ('not' a).z) by BVFUNC_1:def 4;
hence thesis by A2,A1,A3,A4,A5;
end;
hence thesis;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' (a 'imp' b) '&' (b 'imp' (c 'or' a))
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) by Th21;
then
A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' a))=I_el(Y) by
BVFUNC_1:16;
(a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' b)=I_el(Y) by Th38;
then
(a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' b) '&' (b 'imp' (c 'or' a))=
I_el(Y) by A1,th18;
hence thesis by BVFUNC_1:16;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' c)
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) by Th20;
then
A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c))=I_el(Y) by
BVFUNC_1:16;
(a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' c)=I_el(Y) by Th38;
then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c)) '&' (b
'imp' c)=I_el(Y) by A1,th18;
hence thesis by BVFUNC_1:16;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' (a 'imp' (b 'or' c)) '&' (b 'imp' (c 'or' a))
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) by Th21;
then
A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' a))=I_el(Y) by
BVFUNC_1:16;
(a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' c)) by Th19;
then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' c))=I_el(Y) by
BVFUNC_1:16;
then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' c)) '&' (b 'imp' (c
'or' a))=I_el(Y) by A1,th18;
hence thesis by BVFUNC_1:16;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' (c 'or' a))
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) by Th21;
then
A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' a))=I_el(Y) by
BVFUNC_1:16;
(a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) by Th20;
then
(a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c))=I_el(Y) by
BVFUNC_1:16;
then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c)) '&' (b
'imp' (c 'or' a))=I_el(Y) by A1,th18;
hence thesis by BVFUNC_1:16;
end;
theorem
for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b
'imp' c) '<' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' (c 'or' 'not' a))
proof
let a,b,c be Function of Y,BOOLEAN;
(a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' 'not' a)) by Th22;
then
A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' 'not' a))=I_el(Y) by
BVFUNC_1:16;
(a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) by Th20;
then
(a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c))=I_el(Y) by
BVFUNC_1:16;
then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c)) '&' (b
'imp' (c 'or' 'not' a))=I_el(Y) by A1,th18;
hence thesis by BVFUNC_1:16;
end;