deffunc H₁( Element of QC-Sub-WFF ) -> Element of QC-WFF = (the_pred_symbol_of ($1 `1 )) ! (CQC_Subst (Sub_the_arguments_of $1),($1 `2 ));
deffunc H₂( Element of QC-WFF ) -> Element of QC-WFF = 'not' $1;
deffunc H₃( Element of QC-WFF , Element of QC-WFF ) -> Element of QC-WFF = $1 '&' $2;
consider F being Function of QC-Sub-WFF ,QC-WFF such that
A1: for S being Element of QC-Sub-WFF
for d1, d2 being Element of QC-WFF holds
( ( S is Sub_VERUM implies F . S = VERUM ) & ( S is Sub_atomic implies F . S = H₁(S) ) & ( S is Sub_negative & d1 = F . (Sub_the_argument_of S) implies F . S = H₂(d1) ) & ( S is Sub_conjunctive & d1 = F . (Sub_the_left_argument_of S) & d2 = F . (Sub_the_right_argument_of S) implies F . S = H₃(d1,d2) ) & ( S is Sub_universal & d1 = F . (Sub_the_scope_of S) implies F . S = Quant S,d1 ) ) from SUBSTUT1:sch 3();
take F . S ; :: thesis: ex F being Function of QC-Sub-WFF ,QC-WFF st
( F . S = F . S & ( for S' being Element of QC-Sub-WFF holds
( ( S' is Sub_VERUM implies F . S' = VERUM ) & ( S' is Sub_atomic implies F . S' = (the_pred_symbol_of (S' `1 )) ! (CQC_Subst (Sub_the_arguments_of S'),(S' `2 )) ) & ( S' is Sub_negative implies F . S' = 'not' (F . (Sub_the_argument_of S')) ) & ( S' is Sub_conjunctive implies F . S' = (F . (Sub_the_left_argument_of S')) '&' (F . (Sub_the_right_argument_of S')) ) & ( S' is Sub_universal implies F . S' = Quant S',(F . (Sub_the_scope_of S')) ) ) ) )
take F ; :: thesis: ( F . S = F . S & ( for S' being Element of QC-Sub-WFF holds
( ( S' is Sub_VERUM implies F . S' = VERUM ) & ( S' is Sub_atomic implies F . S' = (the_pred_symbol_of (S' `1 )) ! (CQC_Subst (Sub_the_arguments_of S'),(S' `2 )) ) & ( S' is Sub_negative implies F . S' = 'not' (F . (Sub_the_argument_of S')) ) & ( S' is Sub_conjunctive implies F . S' = (F . (Sub_the_left_argument_of S')) '&' (F . (Sub_the_right_argument_of S')) ) & ( S' is Sub_universal implies F . S' = Quant S',(F . (Sub_the_scope_of S')) ) ) ) )
thus F . S = F . S ; :: thesis: for S' being Element of QC-Sub-WFF holds
( ( S' is Sub_VERUM implies F . S' = VERUM ) & ( S' is Sub_atomic implies F . S' = (the_pred_symbol_of (S' `1 )) ! (CQC_Subst (Sub_the_arguments_of S'),(S' `2 )) ) & ( S' is Sub_negative implies F . S' = 'not' (F . (Sub_the_argument_of S')) ) & ( S' is Sub_conjunctive implies F . S' = (F . (Sub_the_left_argument_of S')) '&' (F . (Sub_the_right_argument_of S')) ) & ( S' is Sub_universal implies F . S' = Quant S',(F . (Sub_the_scope_of S')) ) )
thus for S' being Element of QC-Sub-WFF holds
( ( S' is Sub_VERUM implies F . S' = VERUM ) & ( S' is Sub_atomic implies F . S' = (the_pred_symbol_of (S' `1 )) ! (CQC_Subst (Sub_the_arguments_of S'),(S' `2 )) ) & ( S' is Sub_negative implies F . S' = 'not' (F . (Sub_the_argument_of S')) ) & ( S' is Sub_conjunctive implies F . S' = (F . (Sub_the_left_argument_of S')) '&' (F . (Sub_the_right_argument_of S')) ) & ( S' is Sub_universal implies F . S' = Quant S',(F . (Sub_the_scope_of S')) ) ) by A1; :: thesis: verum