let k be Element of NAT ; :: thesis: for A being non empty set
for J being interpretation of A
for P being QC-pred_symbol of k
for ll being CQC-variable_list of
for v being Element of Valuations_in A
for vS, vS1, vS2 being Val_Sub of A st ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let A be non empty set ; :: thesis: for J being interpretation of A
for P being QC-pred_symbol of k
for ll being CQC-variable_list of
for v being Element of Valuations_in A
for vS, vS1, vS2 being Val_Sub of A st ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let J be interpretation of A; :: thesis: for P being QC-pred_symbol of k
for ll being CQC-variable_list of
for v being Element of Valuations_in A
for vS, vS1, vS2 being Val_Sub of A st ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let P be QC-pred_symbol of k; :: thesis: for ll being CQC-variable_list of
for v being Element of Valuations_in A
for vS, vS1, vS2 being Val_Sub of A st ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let ll be CQC-variable_list of ; :: thesis: for v being Element of Valuations_in A
for vS, vS1, vS2 being Val_Sub of A st ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let v be Element of Valuations_in A; :: thesis: for vS, vS1, vS2 being Val_Sub of A st ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let vS, vS1, vS2 be Val_Sub of A; :: thesis: ( ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 implies ( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll ) )
assume A1: ( ( for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 ) ; :: thesis: ( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
A2: for y being bound_QC-variable st y in dom vS1 holds
not y in still_not-bound_in ll

proof

let y be bound_QC-variable; :: thesis: ( y in dom vS1 implies not y in still_not-bound_in ll )
assume y in dom vS1 ; :: thesis: not y in still_not-bound_in ll
then not y in still_not-bound_in (P ! ll) by A1;
hence not y in still_not-bound_in ll by QC_LANG3:9; :: thesis: verum

end;

A3: ( (Valid (P ! ll),J) . (v . vS) = TRUE iff (v . vS) *' ll in J . P ) by VALUAT_1:16;
( (v . ((vS +* vS1) +* vS2)) *' ll in J . P iff (Valid (P ! ll),J) . (v . ((vS +* vS1) +* vS2)) = TRUE ) by VALUAT_1:16;
hence ( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll ) by A1, A2, A3, Th78, VALUAT_1:def 12; :: thesis: verum