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

A2: now

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

end;

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

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

then (B_SUP b,(CompF PA,G)) . z = TRUE by BVFUNC_1:def 20;
then (Ex b,PA,G) . z = TRUE by BVFUNC_2:def 10;
hence ((All a,PA,G) 'or' (Ex b,PA,G)) . z = ((All a,PA,G) . z) 'or' TRUE by BVFUNC_1:def 7
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

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 holds
( not x in EqClass z,(CompF PA,G) or not b . x = TRUE ) ) ) ; :: thesis: ((All a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE

then (B_INF a,(CompF PA,G)) . z = TRUE by BVFUNC_1:def 19;
then (All a,PA,G) . z = TRUE by BVFUNC_2:def 9;
hence ((All a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE 'or' ((Ex b,PA,G) . z) by BVFUNC_1:def 7
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

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

then consider x1 being Element of Y such that
A4: x1 in EqClass z,(CompF PA,G) and
A5: a . x1 <> TRUE ;
A6: b . x1 <> TRUE by A3, A4;
A7: a . x1 = FALSE by A5, XBOOLEAN:def 3;
(a 'or' b) . x1 = (a . x1) 'or' (b . x1) by BVFUNC_1:def 7
.= FALSE 'or' FALSE by A6, A7, XBOOLEAN:def 3
.= FALSE ;
hence ((All a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE by A2, A4; :: thesis: verum

end;

end;