let A be QC-alphabet ; :: thesis: for H being Element of QC-WFF A
for k being Nat
for P being QC-pred_symbol of k,A
for V being QC-variable_list of k,A holds not H is_immediate_constituent_of P ! V
let H be Element of QC-WFF A; :: thesis: for k being Nat
for P being QC-pred_symbol of k,A
for V being QC-variable_list of k,A holds not H is_immediate_constituent_of P ! V
let k be Nat; :: thesis: for P being QC-pred_symbol of k,A
for V being QC-variable_list of k,A holds not H is_immediate_constituent_of P ! V
let P be QC-pred_symbol of k,A; :: thesis: for V being QC-variable_list of k,A holds not H is_immediate_constituent_of P ! V
let V be QC-variable_list of k,A; :: thesis: not H is_immediate_constituent_of P ! V
assume A1: H is_immediate_constituent_of P ! V ; :: thesis: contradiction
A2: P ! V is atomic ;
then A3: ((@ (P ! V)) . 1) `1 <> 3 by QC_LANG1:19;
A4: ((@ (P ! V)) . 1) `1 <> 2 by A2, QC_LANG1:19;
A5: for H1 being Element of QC-WFF A holds
( not P ! V = H '&' H1 & not P ! V = H1 '&' H ) by A4, QC_LANG1:18, QC_LANG1:def 20;
'not' H is negative ;
then A6: ((@ ('not' H)) . 1) `1 = 1 by QC_LANG1:18;
((@ (P ! V)) . 1) `1 <> 1 by A2, QC_LANG1:19;
then consider z being bound_QC-variable of A such that
A7: P ! V = All (z,H) by A1, A6, A5;
All (z,H) is universal ;
hence contradiction by A3, A7, QC_LANG1:18; :: thesis: verum