let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds All (a 'imp' b),PA,G '<' (Ex a,PA,G) 'imp' (Ex b,PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds All (a 'imp' b),PA,G '<' (Ex a,PA,G) 'imp' (Ex b,PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds All (a 'imp' b),PA,G '<' (Ex a,PA,G) 'imp' (Ex b,PA,G)
let PA be a_partition of Y; :: thesis: All (a 'imp' b),PA,G '<' (Ex a,PA,G) 'imp' (Ex b,PA,G)
A1: Ex a,PA,G = B_SUP a,(CompF PA,G) by BVFUNC_2:def 10;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (a 'imp' b),PA,G) . z = TRUE or ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE )
assume A2: (All (a 'imp' b),PA,G) . z = TRUE ; :: thesis: ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

per cases ( 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 ) ) & ( for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not b . x = TRUE ) ) ) ) ;

suppose ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & b . x = TRUE ) ; :: thesis: ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

then (B_SUP b,(CompF PA,G)) . z = TRUE by BVFUNC_1:def 20;
then (Ex b,PA,G) . z = TRUE by BVFUNC_2:def 10;
hence ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = ('not' ((Ex a,PA,G) . z)) 'or' TRUE by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

suppose A3: ( 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 ) ) ) ; :: thesis: ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

then consider x1 being Element of Y such that
A4: ( x1 in EqClass z,(CompF PA,G) & a . x1 = TRUE ) ;
A5: b . x1 <> TRUE by A3, A4;
(a 'imp' b) . x1 = ('not' (a . x1)) 'or' (b . x1) by BVFUNC_1:def 11
.= ('not' TRUE ) 'or' FALSE by A4, A5, XBOOLEAN:def 3
.= FALSE 'or' FALSE by MARGREL1:41
.= FALSE ;
hence ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE by A2, A4, Lm2; :: thesis: verum

end;

suppose A6: ( ( 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 ) ) ) ; :: thesis: ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

thus ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = ('not' ((Ex a,PA,G) . z)) 'or' ((Ex b,PA,G) . z) by BVFUNC_1:def 11
.= ('not' FALSE ) 'or' ((Ex b,PA,G) . z) by A1, A6, BVFUNC_1:def 20
.= TRUE 'or' ((Ex b,PA,G) . z) by MARGREL1:41
.= TRUE by BINARITH:19 ; :: thesis: verum

end;

end;