let Al be QC-alphabet ; :: thesis: for k being Element of NAT
for A being non empty set
for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al holds
( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )
let k be Element of NAT ; :: thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al holds
( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )
let A be non empty set ; :: thesis: for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al holds
( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )
let v be Element of Valuations_in (Al,A); :: thesis: for ll being CQC-variable_list of k,Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al holds
( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )
let ll be CQC-variable_list of k,Al; :: thesis: for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al holds
( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )
let J be interpretation of Al,A; :: thesis: for P being QC-pred_symbol of k,Al holds
( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )
let P be QC-pred_symbol of k,Al; :: thesis: ( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE )

A1: now :: thesis: ( (ll 'in' (J . P)) . v = TRUE implies J,v |= P ! ll )

assume (ll 'in' (J . P)) . v = TRUE ; :: thesis: J,v |= P ! ll
then (Valid ((P ! ll),J)) . v = TRUE by Lm1;
hence J,v |= P ! ll by Def7; :: thesis: verum

end;

now :: thesis: ( J,v |= P ! ll implies (ll 'in' (J . P)) . v = TRUE )

assume J,v |= P ! ll ; :: thesis: (ll 'in' (J . P)) . v = TRUE
then (Valid ((P ! ll),J)) . v = TRUE by Def7;
hence (ll 'in' (J . P)) . v = TRUE by Lm1; :: thesis: verum

end;

hence ( J,v |= P ! ll iff (ll 'in' (J . P)) . v = TRUE ) by A1; :: thesis: verum