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) 'or' (All b,PA,G) '<' a 'or' b
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) 'or' (All b,PA,G) '<' a 'or' b
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds (All a,PA,G) 'or' (All b,PA,G) '<' a 'or' b
let PA be a_partition of Y; :: thesis: (All a,PA,G) 'or' (All b,PA,G) '<' a 'or' b
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((All a,PA,G) 'or' (All b,PA,G)) . z = TRUE or (a 'or' b) . z = TRUE )
assume ((All a,PA,G) 'or' (All b,PA,G)) . z = TRUE ; :: thesis: (a 'or' b) . z = TRUE
then A1: ((All a,PA,G) . z) 'or' ((All b,PA,G) . z) = TRUE by BVFUNC_1:def 7;
A2: ( (All a,PA,G) . z = TRUE or (All a,PA,G) . z = FALSE ) by XBOOLEAN:def 3;
A3: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;

per cases ( (All a,PA,G) . z = TRUE or (All b,PA,G) . z = TRUE ) by A1, A2, BINARITH:7;

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

A5: 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 A4; :: thesis: verum

end;

thus (a 'or' b) . z = (a . z) 'or' (b . z) by BVFUNC_1:def 7
.= TRUE 'or' (b . z) by A3, A5
.= TRUE by BINARITH:19 ; :: thesis: verum

end;

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

A7: 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;
then (All b,PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A6; :: thesis: verum

end;

thus (a 'or' b) . z = (a . z) 'or' (b . z) by BVFUNC_1:def 7
.= (a . z) 'or' TRUE by A3, A7
.= TRUE by BINARITH:19 ; :: thesis: verum

end;

end;