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' (Ex b,PA,G) = Ex (a 'imp' 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' (Ex b,PA,G) = Ex (a 'imp' 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' (Ex b,PA,G) = Ex (a 'imp' b),PA,G
let PA be a_partition of Y; :: thesis: (All a,PA,G) 'imp' (Ex b,PA,G) = Ex (a 'imp' b),PA,G
A1: All a,PA,G = B_INF a,(CompF PA,G) by BVFUNC_2:def 9;
A2: (All a,PA,G) 'imp' (Ex b,PA,G) '<' Ex (a 'imp' b),PA,G

proof

let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE or (Ex (a 'imp' b),PA,G) . z = TRUE )
assume ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE ; :: thesis: (Ex (a 'imp' b),PA,G) . z = TRUE
then A3: ('not' ((All a,PA,G) . z)) 'or' ((Ex b,PA,G) . z) = TRUE by BVFUNC_1:def 11;
A4: ( 'not' ((All a,PA,G) . z) = TRUE or 'not' ((All a,PA,G) . z) = FALSE ) by XBOOLEAN:def 3;

per cases ( 'not' ((All a,PA,G) . z) = TRUE or (Ex b,PA,G) . z = TRUE ) by A3, A4, BINARITH:7;

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

then (All a,PA,G) . z = FALSE by MARGREL1:41;
then consider x1 being Element of Y such that
A5: ( x1 in EqClass z,(CompF PA,G) & a . x1 <> TRUE ) by A1, BVFUNC_1:def 19;
(a 'imp' b) . x1 = ('not' (a . x1)) 'or' (b . x1) by BVFUNC_1:def 11
.= ('not' FALSE ) 'or' (b . x1) by A5, XBOOLEAN:def 3
.= TRUE 'or' (b . x1) by MARGREL1:41
.= TRUE by BINARITH:19 ;
then (B_SUP (a 'imp' b),(CompF PA,G)) . z = TRUE by A5, BVFUNC_1:def 20;
hence (Ex (a 'imp' b),PA,G) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum

end;

suppose A6: (Ex b,PA,G) . z = TRUE ; :: thesis: (Ex (a 'imp' 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 A6; :: thesis: verum

end;

then consider x1 being Element of Y such that
A7: ( x1 in EqClass z,(CompF PA,G) & b . x1 = TRUE ) ;
(a 'imp' b) . x1 = ('not' (a . x1)) 'or' TRUE by A7, BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
then (B_SUP (a 'imp' b),(CompF PA,G)) . z = TRUE by A7, BVFUNC_1:def 20;
hence (Ex (a 'imp' b),PA,G) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum

end;

end;

end;

Ex (a 'imp' b),PA,G '<' (All a,PA,G) 'imp' (Ex b,PA,G)

proof

let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (Ex (a 'imp' b),PA,G) . z = TRUE or ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE )
assume A8: (Ex (a 'imp' b),PA,G) . z = TRUE ; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

now

assume for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not (a 'imp' b) . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (a 'imp' b),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 20;
then (Ex (a 'imp' b),PA,G) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A8; :: thesis: verum

end;

then consider x1 being Element of Y such that
A9: ( x1 in EqClass z,(CompF PA,G) & (a 'imp' b) . x1 = TRUE ) ;
A10: ('not' (a . x1)) 'or' (b . x1) = TRUE by A9, BVFUNC_1:def 11;
A11: ( 'not' (a . x1) = TRUE or 'not' (a . x1) = FALSE ) by XBOOLEAN:def 3;

per cases ( 'not' (a . x1) = TRUE or b . x1 = TRUE ) by A10, A11, BINARITH:7;

suppose 'not' (a . x1) = TRUE ; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

then a . x1 = FALSE by MARGREL1:41;
then (B_INF a,(CompF PA,G)) . z = FALSE by A9, BVFUNC_1:def 19;
hence ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = ('not' FALSE ) 'or' ((Ex b,PA,G) . z) by A1, BVFUNC_1:def 11
.= TRUE 'or' ((Ex b,PA,G) . z) by MARGREL1:41
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

suppose b . x1 = TRUE ; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE

then (B_SUP b,(CompF PA,G)) . z = TRUE by A9, 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;

end;

hence (All a,PA,G) 'imp' (Ex b,PA,G) = Ex (a 'imp' b),PA,G by A2, BVFUNC_1:18; :: thesis: verum