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,PA,G '<' (Ex b,PA,G) 'imp' (Ex (a '&' 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,PA,G '<' (Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds All a,PA,G '<' (Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)
let PA be a_partition of Y; :: thesis: All a,PA,G '<' (Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All a,PA,G) . z = TRUE or ((Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)) . z = TRUE )
assume A1: (All a,PA,G) . z = TRUE ; :: thesis: ((Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)) . z = TRUE

A2: now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) ; :: thesis: contradiction
then (B_INF a,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All a,PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A1; :: thesis: verum

end;

per cases ( (Ex b,PA,G) . z = TRUE or (Ex b,PA,G) . z <> TRUE ) ;

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

now

assume for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not b . x = TRUE ) ; :: thesis: contradiction
then (B_SUP b,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 20;
then (Ex b,PA,G) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A3; :: thesis: verum

end;

then consider x1 being Element of Y such that
A4: ( x1 in EqClass z,(CompF PA,G) & b . x1 = TRUE ) ;
(a '&' b) . x1 = (a . x1) '&' (b . x1) by MARGREL1:def 21
.= TRUE '&' TRUE by A2, A4
.= TRUE ;
then (B_SUP (a '&' b),(CompF PA,G)) . z = TRUE by A4, BVFUNC_1:def 20;
then (Ex (a '&' b),PA,G) . z = TRUE by BVFUNC_2:def 10;
hence ((Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)) . z = ('not' ((Ex b,PA,G) . z)) 'or' TRUE by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

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

then (Ex b,PA,G) . z = FALSE by XBOOLEAN:def 3;
hence ((Ex b,PA,G) 'imp' (Ex (a '&' b),PA,G)) . z = ('not' FALSE ) 'or' ((Ex (a '&' b),PA,G) . z) by BVFUNC_1:def 11
.= TRUE 'or' ((Ex (a '&' b),PA,G) . z) by MARGREL1:41
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

end;