let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for u, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds u 'imp' (Ex a,PA,G) '<' Ex (u 'imp' a),PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for u, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds u 'imp' (Ex a,PA,G) '<' Ex (u 'imp' a),PA,G
let u, a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds u 'imp' (Ex a,PA,G) '<' Ex (u 'imp' a),PA,G
let PA be a_partition of Y; :: thesis: u 'imp' (Ex a,PA,G) '<' Ex (u 'imp' a),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (u 'imp' (Ex a,PA,G)) . z = TRUE or (Ex (u 'imp' a),PA,G) . z = TRUE )
A1: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
assume (u 'imp' (Ex a,PA,G)) . z = TRUE ; :: thesis: (Ex (u 'imp' a),PA,G) . z = TRUE
then A2: ('not' (u . z)) 'or' ((Ex a,PA,G) . z) = TRUE by BVFUNC_1:def 11;
A3: ( (Ex a,PA,G) . z = TRUE or (Ex a,PA,G) . z = FALSE ) by XBOOLEAN:def 3;

now

per cases ( 'not' (u . z) = TRUE or (Ex a,PA,G) . z = TRUE ) by A2, A3, BINARITH:7;

case A4: 'not' (u . z) = TRUE ; :: thesis: (Ex (u 'imp' a),PA,G) . z = TRUE

(u 'imp' a) . z = ('not' (u . z)) 'or' (a . z) by BVFUNC_1:def 11
.= TRUE by A4, BINARITH:19 ;
hence (Ex (u 'imp' a),PA,G) . z = TRUE by A1, BVFUNC_1:def 20; :: thesis: verum

end;

case (Ex a,PA,G) . z = TRUE ; :: thesis: (Ex (u 'imp' a),PA,G) . z = TRUE

then consider x1 being Element of Y such that
A5: x1 in EqClass z,(CompF PA,G) and
A6: a . x1 = TRUE by BVFUNC_1:def 20;
(u 'imp' a) . x1 = ('not' (u . x1)) 'or' (a . x1) by BVFUNC_1:def 11
.= TRUE by A6, BINARITH:19 ;
hence (Ex (u 'imp' a),PA,G) . z = TRUE by A5, BVFUNC_1:def 20; :: thesis: verum

end;

end;

end;

hence (Ex (u 'imp' a),PA,G) . z = TRUE ; :: thesis: verum