let Y be non empty set ; for a, b being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y
let a, b be Function of Y,BOOLEAN; for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y
let G be Subset of (PARTITIONS Y); for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y
let PA be a_partition of Y; ( a 'eqv' b = I_el Y implies (Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y )
assume A1:
a 'eqv' b = I_el Y
; (Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y
then
b 'imp' a = I_el Y
by Th10;
then A2:
('not' b) 'or' a = I_el Y
by Th8;
a 'imp' b = I_el Y
by A1, Th10;
then A3:
('not' a) 'or' b = I_el Y
by Th8;
for z being Element of Y holds ((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z = TRUE
proof
let z be
Element of
Y;
((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z = TRUE
(Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) =
((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) '&' ((Ex (b,PA,G)) 'imp' (Ex (a,PA,G)))
by Th7
.=
(('not' (Ex (a,PA,G))) 'or' (Ex (b,PA,G))) '&' ((Ex (b,PA,G)) 'imp' (Ex (a,PA,G)))
by Th8
.=
(('not' (Ex (a,PA,G))) 'or' (Ex (b,PA,G))) '&' (('not' (Ex (b,PA,G))) 'or' (Ex (a,PA,G)))
by Th8
.=
((('not' (Ex (a,PA,G))) 'or' (Ex (b,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' ((('not' (Ex (a,PA,G))) 'or' (Ex (b,PA,G))) '&' (Ex (a,PA,G)))
by BVFUNC_1:12
.=
((('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' ((Ex (b,PA,G)) '&' ('not' (Ex (b,PA,G))))) 'or' ((('not' (Ex (a,PA,G))) 'or' (Ex (b,PA,G))) '&' (Ex (a,PA,G)))
by BVFUNC_1:12
.=
((('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' ((Ex (b,PA,G)) '&' ('not' (Ex (b,PA,G))))) 'or' ((('not' (Ex (a,PA,G))) '&' (Ex (a,PA,G))) 'or' ((Ex (b,PA,G)) '&' (Ex (a,PA,G))))
by BVFUNC_1:12
.=
((('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' (O_el Y)) 'or' ((('not' (Ex (a,PA,G))) '&' (Ex (a,PA,G))) 'or' ((Ex (b,PA,G)) '&' (Ex (a,PA,G))))
by Th5
.=
((('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' (O_el Y)) 'or' ((O_el Y) 'or' ((Ex (b,PA,G)) '&' (Ex (a,PA,G))))
by Th5
.=
(('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' ((O_el Y) 'or' ((Ex (b,PA,G)) '&' (Ex (a,PA,G))))
by BVFUNC_1:9
.=
(('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) 'or' ((Ex (b,PA,G)) '&' (Ex (a,PA,G)))
by BVFUNC_1:9
;
then A4:
((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z =
((('not' (Ex (a,PA,G))) '&' ('not' (Ex (b,PA,G)))) . z) 'or' (((Ex (b,PA,G)) '&' (Ex (a,PA,G))) . z)
by BVFUNC_1:def 4
.=
((('not' (Ex (a,PA,G))) . z) '&' (('not' (Ex (b,PA,G))) . z)) 'or' (((Ex (b,PA,G)) '&' (Ex (a,PA,G))) . z)
by MARGREL1:def 20
.=
((('not' (Ex (a,PA,G))) . z) '&' (('not' (Ex (b,PA,G))) . z)) 'or' (((Ex (b,PA,G)) . z) '&' ((Ex (a,PA,G)) . z))
by MARGREL1:def 20
.=
(('not' ((Ex (a,PA,G)) . z)) '&' (('not' (Ex (b,PA,G))) . z)) 'or' (((Ex (b,PA,G)) . z) '&' ((Ex (a,PA,G)) . z))
by MARGREL1:def 19
.=
(('not' ((Ex (a,PA,G)) . z)) '&' ('not' ((Ex (b,PA,G)) . z))) 'or' (((Ex (b,PA,G)) . z) '&' ((Ex (a,PA,G)) . z))
by MARGREL1:def 19
;
per cases
( ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) ) or ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) or ( ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) ) or ( ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) )
;
suppose A6:
( ex
x being
Element of
Y st
(
x in EqClass (
z,
(CompF (PA,G))) &
a . x = TRUE ) & ( for
x being
Element of
Y holds
( not
x in EqClass (
z,
(CompF (PA,G))) or not
b . x = TRUE ) ) )
;
((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z = TRUE then consider x1 being
Element of
Y such that A7:
x1 in EqClass (
z,
(CompF (PA,G)))
and A8:
a . x1 = TRUE
;
b . x1 <> TRUE
by A6, A7;
then A9:
b . x1 = FALSE
by XBOOLEAN:def 3;
(('not' a) 'or' b) . x1 =
(('not' a) . x1) 'or' (b . x1)
by BVFUNC_1:def 4
.=
FALSE 'or' FALSE
by A8, A9, MARGREL1:def 19
.=
FALSE
;
hence
((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z = TRUE
by A3, BVFUNC_1:def 11;
verum end; suppose A10:
( ( for
x being
Element of
Y holds
( not
x in EqClass (
z,
(CompF (PA,G))) or not
a . x = TRUE ) ) & ex
x being
Element of
Y st
(
x in EqClass (
z,
(CompF (PA,G))) &
b . x = TRUE ) )
;
((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z = TRUE then consider x1 being
Element of
Y such that A11:
x1 in EqClass (
z,
(CompF (PA,G)))
and A12:
b . x1 = TRUE
;
a . x1 <> TRUE
by A10, A11;
then A13:
a . x1 = FALSE
by XBOOLEAN:def 3;
(('not' b) 'or' a) . x1 =
(('not' b) . x1) 'or' (a . x1)
by BVFUNC_1:def 4
.=
FALSE 'or' FALSE
by A12, A13, MARGREL1:def 19
.=
FALSE
;
hence
((Ex (a,PA,G)) 'eqv' (Ex (b,PA,G))) . z = TRUE
by A2, BVFUNC_1:def 11;
verum end; end;
end;
hence
(Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y
by BVFUNC_1:def 11; verum