let Y be non empty set ; for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
let a, b be Element of Funcs Y,BOOLEAN ; for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(All a,PA,G) 'eqv' (All 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
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
let PA be a_partition of Y; ( a 'eqv' b = I_el Y implies (All a,PA,G) 'eqv' (All b,PA,G) = I_el Y )
assume A1:
a 'eqv' b = I_el Y
; (All a,PA,G) 'eqv' (All 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 ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
proof
let z be
Element of
Y;
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
A4:
now assume that A5:
for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
a . x = TRUE
and A6:
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
b . x = TRUE )
;
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE consider x1 being
Element of
Y such that A7:
x1 in EqClass z,
(CompF PA,G)
and A8:
b . x1 <> TRUE
by A6;
A9:
a . x1 = TRUE
by A5, A7;
A10:
b . x1 = FALSE
by A8, XBOOLEAN:def 3;
(('not' a) 'or' b) . x1 =
(('not' a) . x1) 'or' (b . x1)
by BVFUNC_1:def 7
.=
FALSE 'or' FALSE
by A10, A9, MARGREL1:def 20
.=
FALSE
;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A3, BVFUNC_1:def 14;
verum end;
A11:
now assume that A12:
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE )
and A13:
for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
b . x = TRUE
;
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE consider x1 being
Element of
Y such that A14:
x1 in EqClass z,
(CompF PA,G)
and A15:
a . x1 <> TRUE
by A12;
A16:
b . x1 = TRUE
by A13, A14;
A17:
a . x1 = FALSE
by A15, XBOOLEAN:def 3;
(('not' b) 'or' a) . x1 =
(('not' b) . x1) 'or' (a . x1)
by BVFUNC_1:def 7
.=
FALSE 'or' FALSE
by A17, A16, MARGREL1:def 20
.=
FALSE
;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A2, BVFUNC_1:def 14;
verum end;
(All a,PA,G) 'eqv' (All b,PA,G) =
((All a,PA,G) 'imp' (All b,PA,G)) '&' ((All b,PA,G) 'imp' (All a,PA,G))
by Th7
.=
(('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' ((All b,PA,G) 'imp' (All a,PA,G))
by Th8
.=
(('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' (('not' (All b,PA,G)) 'or' (All a,PA,G))
by Th8
.=
((('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' (All a,PA,G))
by BVFUNC_1:15
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((All b,PA,G) '&' ('not' (All b,PA,G)))) 'or' ((('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' (All a,PA,G))
by BVFUNC_1:15
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((All b,PA,G) '&' ('not' (All b,PA,G)))) 'or' ((('not' (All a,PA,G)) '&' (All a,PA,G)) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by BVFUNC_1:15
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' (O_el Y)) 'or' ((('not' (All a,PA,G)) '&' (All a,PA,G)) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by Th5
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' (O_el Y)) 'or' ((O_el Y) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by Th5
.=
(('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((O_el Y) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by BVFUNC_1:12
.=
(('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((All b,PA,G) '&' (All a,PA,G))
by BVFUNC_1:12
;
then A18:
((All a,PA,G) 'eqv' (All b,PA,G)) . z =
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) . z) 'or' (((All b,PA,G) '&' (All a,PA,G)) . z)
by BVFUNC_1:def 7
.=
((('not' (All a,PA,G)) . z) '&' (('not' (All b,PA,G)) . z)) 'or' (((All b,PA,G) '&' (All a,PA,G)) . z)
by MARGREL1:def 21
.=
((('not' (All a,PA,G)) . z) '&' (('not' (All b,PA,G)) . z)) 'or' (((All b,PA,G) . z) '&' ((All a,PA,G) . z))
by MARGREL1:def 21
.=
(('not' ((All a,PA,G) . z)) '&' (('not' (All b,PA,G)) . z)) 'or' (((All b,PA,G) . z) '&' ((All a,PA,G) . z))
by MARGREL1:def 20
.=
(('not' ((All a,PA,G) . z)) '&' ('not' ((All b,PA,G) . z))) 'or' (((All b,PA,G) . z) '&' ((All a,PA,G) . z))
by MARGREL1:def 20
;
A19:
now assume that A20:
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE )
and A21:
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
b . x = TRUE )
;
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
(B_INF b,(CompF PA,G)) . z = FALSE
by A21, BVFUNC_1:def 19;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A18, A20, BVFUNC_1:def 19;
verum end;
now assume that A22:
for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
a . x = TRUE
and A23:
for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
b . x = TRUE
;
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
(B_INF b,(CompF PA,G)) . z = TRUE
by A23, BVFUNC_1:def 19;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A18, A22, BVFUNC_1:def 19;
verum end;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A4, A11, A19;
verum
end;
hence
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
by BVFUNC_1:def 14; verum