let S be Element of QC-Sub-WFF ; :: thesis: ( S is Sub_negative implies CQC_Sub S = 'not' (CQC_Sub (Sub_the_argument_of S)) )
consider F being Function of QC-Sub-WFF ,QC-WFF such that
A1: CQC_Sub S = F . S and
A2: for S9 being Element of QC-Sub-WFF holds
( ( S9 is Sub_VERUM implies F . S9 = VERUM ) & ( S9 is Sub_atomic implies F . S9 = (the_pred_symbol_of (S9 `1 )) ! (CQC_Subst (Sub_the_arguments_of S9),(S9 `2 )) ) & ( S9 is Sub_negative implies F . S9 = 'not' (F . (Sub_the_argument_of S9)) ) & ( S9 is Sub_conjunctive implies F . S9 = (F . (Sub_the_left_argument_of S9)) '&' (F . (Sub_the_right_argument_of S9)) ) & ( S9 is Sub_universal implies F . S9 = Quant S9,(F . (Sub_the_scope_of S9)) ) ) by Def38;
consider G being Function of QC-Sub-WFF ,QC-WFF such that
A3: CQC_Sub (Sub_the_argument_of S) = G . (Sub_the_argument_of S) and
A4: for S9 being Element of QC-Sub-WFF holds
( ( S9 is Sub_VERUM implies G . S9 = VERUM ) & ( S9 is Sub_atomic implies G . S9 = (the_pred_symbol_of (S9 `1 )) ! (CQC_Subst (Sub_the_arguments_of S9),(S9 `2 )) ) & ( S9 is Sub_negative implies G . S9 = 'not' (G . (Sub_the_argument_of S9)) ) & ( S9 is Sub_conjunctive implies G . S9 = (G . (Sub_the_left_argument_of S9)) '&' (G . (Sub_the_right_argument_of S9)) ) & ( S9 is Sub_universal implies G . S9 = Quant S9,(G . (Sub_the_scope_of S9)) ) ) by Def38;
F = G by A2, A4, Lm5;
hence ( S is Sub_negative implies CQC_Sub S = 'not' (CQC_Sub (Sub_the_argument_of S)) ) by A1, A2, A3; :: thesis: verum