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) '<' All (a 'or' 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) 'or' (All b,PA,G) '<' All (a 'or' 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) 'or' (All b,PA,G) '<' All (a 'or' b),PA,G
let PA be a_partition of Y; :: thesis: (All a,PA,G) 'or' (All b,PA,G) '<' All (a 'or' b),PA,G
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 (All (a 'or' b),PA,G) . z = TRUE )
assume ((All a,PA,G) 'or' (All b,PA,G)) . z = TRUE ; :: thesis: (All (a 'or' b),PA,G) . z = TRUE
then A1: ((All a,PA,G) . z) 'or' ((All b,PA,G) . z) = TRUE by BVFUNC_1:def 7;
A2: ( (All b,PA,G) . z = TRUE or (All b,PA,G) . z = FALSE ) by XBOOLEAN:def 3;

now

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

case A3: (All a,PA,G) . z = TRUE ; :: thesis: (All (a 'or' b),PA,G) . z = TRUE

for x being Element of Y st x in EqClass z,(CompF PA,G) holds
(a 'or' b) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass z,(CompF PA,G) implies (a 'or' b) . x = TRUE )
assume A4: x in EqClass z,(CompF PA,G) ; :: thesis: (a 'or' b) . x = TRUE
(a 'or' b) . x = (a . x) 'or' (b . x) by BVFUNC_1:def 7
.= TRUE 'or' (b . x) by A3, A4, BVFUNC_1:def 19
.= TRUE by BINARITH:19 ;
hence (a 'or' b) . x = TRUE ; :: thesis: verum

end;

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

end;

case A5: (All b,PA,G) . z = TRUE ; :: thesis: (All (a 'or' b),PA,G) . z = TRUE

for x being Element of Y st x in EqClass z,(CompF PA,G) holds
(a 'or' b) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass z,(CompF PA,G) implies (a 'or' b) . x = TRUE )
assume A6: x in EqClass z,(CompF PA,G) ; :: thesis: (a 'or' b) . x = TRUE
(a 'or' b) . x = (a . x) 'or' (b . x) by BVFUNC_1:def 7
.= (a . x) 'or' TRUE by A5, A6, BVFUNC_1:def 19
.= TRUE by BINARITH:19 ;
hence (a 'or' b) . x = TRUE ; :: thesis: verum

end;

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

end;

end;

end;

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