let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds Ex (a,PA,G) is_dependent_of CompF (PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds Ex (a,PA,G) is_dependent_of CompF (PA,G)
let a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds Ex (a,PA,G) is_dependent_of CompF (PA,G)
let PA be a_partition of Y; :: thesis: Ex (a,PA,G) is_dependent_of CompF (PA,G)
let F be set ; :: according to BVFUNC_1:def 15 :: thesis: ( not F in CompF (PA,G) or for b₁, b₂ being set holds
( not b₁ in F or not b₂ in F or (Ex (a,PA,G)) . b₁ = (Ex (a,PA,G)) . b₂ ) )
assume A1: F in CompF (PA,G) ; :: thesis: for b₁, b₂ being set holds
( not b₁ in F or not b₂ in F or (Ex (a,PA,G)) . b₁ = (Ex (a,PA,G)) . b₂ )
thus for x1, x2 being set st x1 in F & x2 in F holds
(Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2 :: thesis: verum

proof

let x1, x2 be set ; :: thesis: ( x1 in F & x2 in F implies (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2 )
assume A2: ( x1 in F & x2 in F ) ; :: thesis: (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2
then reconsider x1 = x1, x2 = x2 as Element of Y by A1;
A3: x2 in EqClass (x2,(CompF (PA,G))) by EQREL_1:def 6;
( F = EqClass (x2,(CompF (PA,G))) or F misses EqClass (x2,(CompF (PA,G))) ) by A1, EQREL_1:def 4;
then A4: EqClass (x1,(CompF (PA,G))) = EqClass (x2,(CompF (PA,G))) by A2, A3, EQREL_1:def 6, XBOOLE_0:3;

per cases ( ( ex x being Element of Y st
( x in EqClass (x1,(CompF (PA,G))) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (x2,(CompF (PA,G))) & a . x = TRUE ) ) or ( ex x being Element of Y st
( x in EqClass (x1,(CompF (PA,G))) & a . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (x2,(CompF (PA,G))) or not a . x = TRUE ) ) ) or ( ( for x being Element of Y holds
( not x in EqClass (x1,(CompF (PA,G))) or not a . x = TRUE ) ) & ex x being Element of Y st
( x in EqClass (x2,(CompF (PA,G))) & a . x = TRUE ) ) or ( ( for x being Element of Y holds
( not x in EqClass (x1,(CompF (PA,G))) or not a . x = TRUE ) ) & ( for x being Element of Y holds
( not x in EqClass (x2,(CompF (PA,G))) or not a . x = TRUE ) ) ) ) ;

suppose A5: ( ex x being Element of Y st
( x in EqClass (x1,(CompF (PA,G))) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (x2,(CompF (PA,G))) & a . x = TRUE ) ) ; :: thesis: (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2

then (B_SUP (a,(CompF (PA,G)))) . x1 = TRUE by BVFUNC_1:def 17;
hence (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2 by A5, BVFUNC_1:def 17; :: thesis: verum

end;

suppose ( ex x being Element of Y st
( x in EqClass (x1,(CompF (PA,G))) & a . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (x2,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2

hence (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2 by A4; :: thesis: verum

end;

suppose ( ( for x being Element of Y holds
( not x in EqClass (x1,(CompF (PA,G))) or not a . x = TRUE ) ) & ex x being Element of Y st
( x in EqClass (x2,(CompF (PA,G))) & a . x = TRUE ) ) ; :: thesis: (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2

hence (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2 by A4; :: thesis: verum

end;

suppose A6: ( ( for x being Element of Y holds
( not x in EqClass (x1,(CompF (PA,G))) or not a . x = TRUE ) ) & ( for x being Element of Y holds
( not x in EqClass (x2,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2

then (B_SUP (a,(CompF (PA,G)))) . x1 = FALSE by BVFUNC_1:def 17;
hence (Ex (a,PA,G)) . x1 = (Ex (a,PA,G)) . x2 by A6, BVFUNC_1:def 17; :: thesis: verum

end;

end;

end;