let Al be QC-alphabet ; :: thesis: for k being 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 p being Element of CQC-WFF Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
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 k be 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 p being Element of CQC-WFF Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
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 (Al,A)
for ll being CQC-variable_list of k,Al
for p being Element of CQC-WFF Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
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 (Al,A); :: thesis: for ll being CQC-variable_list of k,Al
for p being Element of CQC-WFF Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
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 k,Al; :: thesis: for p being Element of CQC-WFF Al
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
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 Al; :: thesis: for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
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 Al,A; :: thesis: for P being QC-pred_symbol of k,Al
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,Al; :: 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 that
A1: p = P ! ll and
A2: r = J . P ; :: thesis: ( not v *' ll in r iff (Valid (p,J)) . v = FALSE )

A3: now :: thesis: ( (Valid (p,J)) . v = FALSE implies not v *' ll in r )

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 A2, Def4; :: thesis: verum

end;

now :: thesis: ( not v *' ll in r implies (Valid (p,J)) . v = FALSE )

assume not v *' ll in r ; :: thesis: (Valid (p,J)) . v = FALSE
then (ll 'in' (J . P)) . v = FALSE by A2, Def4;
hence (Valid (p,J)) . v = FALSE by A1, Lm1; :: thesis: verum

end;

hence ( not v *' ll in r iff (Valid (p,J)) . v = FALSE ) by A3; :: thesis: verum