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) 'imp' (All b,PA,G) '<' (All 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,PA,G) 'imp' (All b,PA,G) '<' (All 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,PA,G) 'imp' (All b,PA,G) '<' (All a,PA,G) 'imp' (Ex b,PA,G)
let PA be a_partition of Y; :: thesis: (All a,PA,G) 'imp' (All b,PA,G) '<' (All a,PA,G) 'imp' (Ex b,PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE or ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE )
A1: ( 'not' ((All a,PA,G) . z) = TRUE or 'not' ((All a,PA,G) . z) = FALSE ) by XBOOLEAN:def 3;
A2: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
assume ((All a,PA,G) 'imp' (All b,PA,G)) . z = TRUE ; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE
then A3: ('not' ((All a,PA,G) . z)) 'or' ((All b,PA,G) . z) = TRUE by BVFUNC_1:def 11;

per cases ( 'not' ((All a,PA,G) . z) = TRUE or (All b,PA,G) . z = TRUE ) by A3, A1, BINARITH:7;

suppose 'not' ((All a,PA,G) . z) = TRUE ; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

hence ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE 'or' ((Ex b,PA,G) . z) by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

suppose A4: (All b,PA,G) . z = TRUE ; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ; :: thesis: contradiction
then (B_INF b,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
hence contradiction by A4, BVFUNC_2:def 9; :: thesis: verum

end;

then b . z = TRUE by A2;
then (B_SUP b,(CompF PA,G)) . z = TRUE by A2, BVFUNC_1:def 20;
then (Ex b,PA,G) . z = TRUE by BVFUNC_2:def 10;
hence ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = ('not' ((All a,PA,G) . z)) 'or' TRUE by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

end;