let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y holds All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
let PA be a_partition of Y; :: thesis: All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
consider k3 being Function such that
A1: All (a 'eqv' b),PA,G = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G) = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for z being Element of Y holds (All (a 'eqv' b),PA,G) . z = ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z

proof

let z be Element of Y; :: thesis: (All (a 'eqv' b),PA,G) . z = ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z
(All (a 'eqv' b),PA,G) . z = (All ((a 'imp' b) '&' (b 'imp' a)),PA,G) . z by Th7
.= ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z by BVFUNC_1:42 ;
hence (All (a 'eqv' b),PA,G) . z = ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G) by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum