let k be Element of NAT ; :: thesis: for P being QC-pred_symbol of k
for ll being CQC-variable_list of k holds still_not-bound_in (P ! ll) c= Bound_Vars (P ! ll)
let P be QC-pred_symbol of k; :: thesis: for ll being CQC-variable_list of k holds still_not-bound_in (P ! ll) c= Bound_Vars (P ! ll)
let ll be CQC-variable_list of k; :: thesis: still_not-bound_in (P ! ll) c= Bound_Vars (P ! ll)
set l1 = the_arguments_of (P ! ll);
A1: P ! ll is atomic by QC_LANG1:def 16;
then consider n being Element of NAT , P9 being QC-pred_symbol of n, ll9 being QC-variable_list of n such that
A2: the_arguments_of (P ! ll) = ll9 and
A3: P ! ll = P9 ! ll9 by QC_LANG1:def 21;
Bound_Vars (P ! ll) = Bound_Vars (the_arguments_of (P ! ll)) by A1, SUBSTUT1:3;
then A4: Bound_Vars (P ! ll) = { ((the_arguments_of (P ! ll)) . i) where i is Element of NAT : ( 1 <= i & i <= len (the_arguments_of (P ! ll)) & (the_arguments_of (P ! ll)) . i in bound_QC-variables ) } by SUBSTUT1:def 7;
still_not-bound_in (P ! ll) = still_not-bound_in ll by QC_LANG3:5;
then A5: still_not-bound_in (P ! ll) = variables_in (ll,bound_QC-variables) by QC_LANG3:2;
A6: ( (<*P9*> ^ ll9) . 1 = P9 & (<*P*> ^ ll) . 1 = P ) by FINSEQ_1:41;
( P ! ll = <*P*> ^ ll & P9 ! ll9 = <*P9*> ^ ll9 ) by QC_LANG1:6;
then ll9 = ll by A3, A6, FINSEQ_1:33;
hence still_not-bound_in (P ! ll) c= Bound_Vars (P ! ll) by A4, A5, A2, QC_LANG3:def 1; :: thesis: verum