let k be Element of NAT ; :: thesis: for A being non empty set
for v being Element of Valuations_in A
for ll being CQC-variable_list of
for p being Element of CQC-WFF
for J being interpretation of A
for P being QC-pred_symbol of k
for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let A be non empty set ; :: thesis: for v being Element of Valuations_in A
for ll being CQC-variable_list of
for p being Element of CQC-WFF
for J being interpretation of A
for P being QC-pred_symbol of k
for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let v be Element of Valuations_in A; :: thesis: for ll being CQC-variable_list of
for p being Element of CQC-WFF
for J being interpretation of A
for P being QC-pred_symbol of k
for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let ll be CQC-variable_list of ; :: thesis: for p being Element of CQC-WFF
for J being interpretation of A
for P being QC-pred_symbol of k
for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let p be Element of CQC-WFF ; :: thesis: for J being interpretation of A
for P being QC-pred_symbol of k
for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let J be interpretation of A; :: thesis: for P being QC-pred_symbol of k
for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let P be QC-pred_symbol of k; :: thesis: for r being Element of relations_on A st p = P ! ll & r = J . P holds
( not v *' ll in r iff (Valid p,J) . v = FALSE )
let r be Element of relations_on A; :: thesis: ( p = P ! ll & r = J . P implies ( not v *' ll in r iff (Valid p,J) . v = FALSE ) )
assume A1: ( p = P ! ll & r = J . P ) ; :: thesis: ( not v *' ll in r iff (Valid p,J) . v = FALSE )

A2: now

assume not v *' ll in r ; :: thesis: (Valid p,J) . v = FALSE
then (ll 'in' (J . P)) . v = FALSE by A1, Def9;
hence (Valid p,J) . v = FALSE by A1, Lm1; :: thesis: verum

end;

now

assume (Valid p,J) . v = FALSE ; :: thesis: not v *' ll in r
then (ll 'in' (J . P)) . v <> TRUE by A1, Lm1;
hence not v *' ll in r by A1, Def9; :: thesis: verum

end;

hence ( not v *' ll in r iff (Valid p,J) . v = FALSE ) by A2; :: thesis: verum