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_1:39 ;
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_1:2; :: thesis: verum