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 '<' (All a,PA,G) 'imp' (All 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 '<' (All a,PA,G) 'imp' (All 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 '<' (All a,PA,G) 'imp' (All b,PA,G)
let PA be a_partition of Y; :: thesis: All (a 'imp' b),PA,G '<' (All a,PA,G) 'imp' (All b,PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (a 'imp' b),PA,G) . z = TRUE or ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE )
assume A1: (All (a 'imp' b),PA,G) . z = TRUE ; :: thesis: ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE
A2: ((All a,PA,G) 'imp' (All b,PA,G)) . z = ('not' ((All a,PA,G) . z)) 'or' ((All b,PA,G) . z) by BVFUNC_1:def 11;

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

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

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

end;

suppose A3: ( ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
a . x = TRUE ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ) ; :: thesis: ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE

then consider x1 being Element of Y such that
A4: x1 in EqClass z,(CompF PA,G) and
A5: b . x1 <> TRUE ;
A6: a . 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 A5, A6, XBOOLEAN:def 3
.= FALSE 'or' FALSE by MARGREL1:41
.= FALSE ;
hence ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE by A1, A4, BVFUNC_1:def 19; :: thesis: verum

end;

suppose ( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) & ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
b . x = TRUE ) ) ; :: thesis: ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE

then ((All a,PA,G) 'imp' (All b,PA,G)) . z = ('not' ((All a,PA,G) . z)) 'or' TRUE by A2, BVFUNC_1:def 19
.= TRUE by BINARITH:19 ;
hence ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE ; :: thesis: verum

end;

suppose ( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ) ; :: thesis: ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE

then ((All a,PA,G) 'imp' (All b,PA,G)) . z = ('not' FALSE ) 'or' ((All b,PA,G) . z) by A2, BVFUNC_1:def 19
.= TRUE 'or' ((All b,PA,G) . z) by MARGREL1:41
.= TRUE by BINARITH:19 ;
hence ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE ; :: thesis: verum

end;

end;