let S be non empty non void bool-correct 4,1 integer BoolSignature ; :: thesis: for X being non-empty ManySortedSet of the carrier of S
for T being b₁,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer VarMSAlgebra over S
for C being bool-correct 4,1 integer image of T
for u being ManySortedFunction of FreeGen T, the Sorts of C
for a being boolean object
for t being Element of T, the bool-sort of S holds
( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )
let X be non-empty ManySortedSet of the carrier of S; :: thesis: for T being X,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer VarMSAlgebra over S
for C being bool-correct 4,1 integer image of T
for u being ManySortedFunction of FreeGen T, the Sorts of C
for a being boolean object
for t being Element of T, the bool-sort of S holds
( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )
let T be X,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer VarMSAlgebra over S; :: thesis: for C being bool-correct 4,1 integer image of T
for u being ManySortedFunction of FreeGen T, the Sorts of C
for a being boolean object
for t being Element of T, the bool-sort of S holds
( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )
let C be bool-correct 4,1 integer image of T; :: thesis: for u being ManySortedFunction of FreeGen T, the Sorts of C
for a being boolean object
for t being Element of T, the bool-sort of S holds
( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )
let u be ManySortedFunction of FreeGen T, the Sorts of C; :: thesis: for a being boolean object
for t being Element of T, the bool-sort of S holds
( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )
let a be boolean object ; :: thesis: for t being Element of T, the bool-sort of S holds
( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )
let t be Element of T, the bool-sort of S; :: thesis: ( (\not t) value_at (C,u) = 'not' a iff t value_at (C,u) = a )

hereby :: thesis: ( t value_at (C,u) = a implies (\not t) value_at (C,u) = 'not' a )

assume (\not t) value_at (C,u) = 'not' a ; :: thesis: t value_at (C,u) = a
then \not (t value_at (C,u)) = 'not' a by Th47;
then 'not' (t value_at (C,u)) = 'not' a by AOFA_A00:def 32;
hence t value_at (C,u) = a ; :: thesis: verum

end;

assume A1: t value_at (C,u) = a ; :: thesis: (\not t) value_at (C,u) = 'not' a
\not (t value_at (C,u)) = (\not t) value_at (C,u) by Th47;
hence (\not t) value_at (C,u) = 'not' a by A1, AOFA_A00:def 32; :: thesis: verum