Article SUBLEMMA, MML version 4.99.1005
:: SUBLEMMA:th 1
theorem
for b1, b2, b3, b4, b5 being Relation-like Function-like set
st proj1 b4 c= proj1 b3 & proj1 b5 c= proj1 b3
holds (b1 +* b2) +* b3 = ((b1 +* b4) +* (b2 +* b5)) +* b3;
:: SUBLEMMA:th 2
theorem
for b1 being Element of bound_QC-variables
for b2 being Relation-like Function-like set
st b1 in proj1 b2
holds (b2 | ((proj1 b2) \ {b1})) +* (b1 .--> (b2 . b1)) = b2;
:: SUBLEMMA:modenot 1
definition
let a1 be non empty set;
mode Val_Sub of a1 is Function-like Relation of bound_QC-variables,a1;
end;
:: SUBLEMMA:funcnot 1 => SUBLEMMA:func 1
definition
let a1 be non empty set;
let a2 be Element of Valuations_in a1;
let a3 be Function-like Relation of bound_QC-variables,a1;
func A2 . A3 -> Element of Valuations_in a1 equals
a2 +* a3;
end;
:: SUBLEMMA:def 1
theorem
for b1 being non empty set
for b2 being Element of Valuations_in b1
for b3 being Function-like Relation of bound_QC-variables,b1 holds
b2 . b3 = b2 +* b3;
:: SUBLEMMA:funcnot 2 => SUBLEMMA:func 2
definition
let a1 be Element of CQC-Sub-WFF;
redefine func a1 `1 -> Element of CQC-WFF;
end;
:: SUBLEMMA:funcnot 3 => SUBLEMMA:func 3
definition
let a1 be Element of CQC-Sub-WFF;
let a2 be non empty set;
let a3 be Element of Valuations_in a2;
func Val_S(A3,A1) -> Function-like Relation of bound_QC-variables,a2 equals
(@ (a1 `2)) * a3;
end;
:: SUBLEMMA:def 2
theorem
for b1 being Element of CQC-Sub-WFF
for b2 being non empty set
for b3 being Element of Valuations_in b2 holds
Val_S(b3,b1) = (@ (b1 `2)) * b3;
:: SUBLEMMA:th 3
theorem
for b1 being Element of CQC-Sub-WFF
st b1 is Sub_VERUM
holds CQC_Sub b1 = VERUM;
:: SUBLEMMA:prednot 1 => SUBLEMMA:pred 1
definition
let a1 be Element of CQC-Sub-WFF;
let a2 be non empty set;
let a3 be Element of Valuations_in a2;
let a4 be interpretation of a2;
pred A4,A3 |= A1 means
a4,a3 |= a1 `1;
end;
:: SUBLEMMA:dfs 3
definiens
let a1 be Element of CQC-Sub-WFF;
let a2 be non empty set;
let a3 be Element of Valuations_in a2;
let a4 be interpretation of a2;
To prove
a4,a3 |= a1
it is sufficient to prove
thus a4,a3 |= a1 `1;
:: SUBLEMMA:def 3
theorem
for b1 being Element of CQC-Sub-WFF
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being interpretation of b2 holds
b4,b3 |= b1
iff
b4,b3 |= b1 `1;
:: SUBLEMMA:th 4
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of CQC-Sub-WFF
st b3 is Sub_VERUM
for b4 being Element of Valuations_in b1 holds
b2,b4 |= CQC_Sub b3
iff
b2,b4 . Val_S(b4,b3) |= b3;
:: SUBLEMMA:th 5
theorem
for b1, b2 being Element of NAT
for b3 being CQC-variable_list-like QC-variable_list of b1
st b2 in dom b3
holds b3 . b2 is Element of bound_QC-variables;
:: SUBLEMMA:th 6
theorem
for b1 being Element of CQC-Sub-WFF
st b1 is Sub_atomic
holds CQC_Sub b1 = (the_pred_symbol_of (b1 `1)) ! CQC_Subst(Sub_the_arguments_of b1,b1 `2);
:: SUBLEMMA:th 7
theorem
for b1 being Element of NAT
for b2, b3 being Element of b1 -ary_QC-pred_symbols
for b4, b5 being CQC-variable_list-like QC-variable_list of b1
for b6, b7 being Element of vSUB
st Sub_the_arguments_of Sub_P(b2,b4,b6) = Sub_the_arguments_of Sub_P(b3,b5,b7)
holds b4 = b5;
:: SUBLEMMA:funcnot 4 => SUBLEMMA:func 4
definition
let a1 be Element of NAT;
let a2 be Element of a1 -ary_QC-pred_symbols;
let a3 be CQC-variable_list-like QC-variable_list of a1;
let a4 be Element of vSUB;
redefine func Sub_P(a2,a3,a4) -> Element of CQC-Sub-WFF;
end;
:: SUBLEMMA:th 9
theorem
for b1 being Element of NAT
for b2 being Element of b1 -ary_QC-pred_symbols
for b3 being CQC-variable_list-like QC-variable_list of b1
for b4 being Element of vSUB holds
CQC_Sub Sub_P(b2,b3,b4) = b2 ! CQC_Subst(b3,b4);
:: SUBLEMMA:th 10
theorem
for b1 being Element of NAT
for b2 being Element of b1 -ary_QC-pred_symbols
for b3 being CQC-variable_list-like QC-variable_list of b1
for b4 being Element of vSUB holds
b2 ! CQC_Subst(b3,b4) is Element of CQC-WFF;
:: SUBLEMMA:th 11
theorem
for b1 being Element of NAT
for b2 being CQC-variable_list-like QC-variable_list of b1
for b3 being Element of vSUB holds
CQC_Subst(b2,b3) is CQC-variable_list-like QC-variable_list of b1;
:: SUBLEMMA:funcnot 5 => SUBLEMMA:func 5
definition
let a1 be Element of NAT;
let a2 be CQC-variable_list-like QC-variable_list of a1;
let a3 be Element of vSUB;
redefine func CQC_Subst(a2,a3) -> CQC-variable_list-like QC-variable_list of a1;
end;
:: SUBLEMMA:th 12
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
st not b1 in proj1 (b4 `2)
holds (b3 . Val_S(b3,b4)) . b1 = b3 . b1;
:: SUBLEMMA:th 13
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
st b1 in proj1 (b4 `2)
holds (b3 . Val_S(b3,b4)) . b1 = (Val_S(b3,b4)) . b1;
:: SUBLEMMA:th 14
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of b1 -ary_QC-pred_symbols
for b5 being CQC-variable_list-like QC-variable_list of b1
for b6 being Element of vSUB holds
(b3 . Val_S(b3,Sub_P(b4,b5,b6))) *' b5 = b3 *' CQC_Subst(b5,b6);
:: SUBLEMMA:th 15
theorem
for b1 being Element of NAT
for b2 being Element of b1 -ary_QC-pred_symbols
for b3 being CQC-variable_list-like QC-variable_list of b1
for b4 being Element of vSUB holds
(Sub_P(b2,b3,b4)) `1 = b2 ! b3;
:: SUBLEMMA:th 16
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of b1 -ary_QC-pred_symbols
for b5 being CQC-variable_list-like QC-variable_list of b1
for b6 being Element of vSUB
for b7 being Element of Valuations_in b2 holds
b3,b7 |= CQC_Sub Sub_P(b4,b5,b6)
iff
b3,b7 . Val_S(b7,Sub_P(b4,b5,b6)) |= Sub_P(b4,b5,b6);
:: SUBLEMMA:th 17
theorem
for b1 being Element of CQC-Sub-WFF holds
(Sub_not b1) `1 = 'not' (b1 `1) & (Sub_not b1) `2 = b1 `2;
:: SUBLEMMA:funcnot 6 => SUBLEMMA:func 6
definition
let a1 be Element of CQC-Sub-WFF;
redefine func Sub_not a1 -> Element of CQC-Sub-WFF;
end;
:: SUBLEMMA:th 18
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of Valuations_in b1
for b4 being Element of CQC-Sub-WFF holds
not b2,b3 . Val_S(b3,b4) |= b4
iff
b2,b3 . Val_S(b3,b4) |= Sub_not b4;
:: SUBLEMMA:th 19
theorem
for b1 being non empty set
for b2 being Element of Valuations_in b1
for b3 being Element of CQC-Sub-WFF holds
Val_S(b2,b3) = Val_S(b2,Sub_not b3);
:: SUBLEMMA:th 20
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of CQC-Sub-WFF
st for b4 being Element of Valuations_in b1 holds
b2,b4 |= CQC_Sub b3
iff
b2,b4 . Val_S(b4,b3) |= b3
for b4 being Element of Valuations_in b1 holds
b2,b4 |= CQC_Sub Sub_not b3
iff
b2,b4 . Val_S(b4,Sub_not b3) |= Sub_not b3;
:: SUBLEMMA:funcnot 7 => SUBLEMMA:func 7
definition
let a1, a2 be Element of CQC-Sub-WFF;
assume a1 `2 = a2 `2;
func CQCSub_&(A1,A2) -> Element of CQC-Sub-WFF equals
Sub_&(a1,a2);
end;
:: SUBLEMMA:def 4
theorem
for b1, b2 being Element of CQC-Sub-WFF
st b1 `2 = b2 `2
holds CQCSub_&(b1,b2) = Sub_&(b1,b2);
:: SUBLEMMA:th 21
theorem
for b1, b2 being Element of CQC-Sub-WFF
st b1 `2 = b2 `2
holds (CQCSub_&(b1,b2)) `1 = b1 `1 '&' (b2 `1) &
(CQCSub_&(b1,b2)) `2 = b1 `2;
:: SUBLEMMA:th 22
theorem
for b1, b2 being Element of CQC-Sub-WFF
st b1 `2 = b2 `2
holds (CQCSub_&(b1,b2)) `2 = b1 `2;
:: SUBLEMMA:th 23
theorem
for b1 being non empty set
for b2 being Element of Valuations_in b1
for b3, b4 being Element of CQC-Sub-WFF
st b3 `2 = b4 `2
holds Val_S(b2,b3) = Val_S(b2,CQCSub_&(b3,b4)) & Val_S(b2,b4) = Val_S(b2,CQCSub_&(b3,b4));
:: SUBLEMMA:th 24
theorem
for b1, b2 being Element of CQC-Sub-WFF
st b1 `2 = b2 `2
holds CQC_Sub CQCSub_&(b1,b2) = (CQC_Sub b1) '&' CQC_Sub b2;
:: SUBLEMMA:th 25
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of Valuations_in b1
for b4, b5 being Element of CQC-Sub-WFF
st b4 `2 = b5 `2
holds b2,b3 . Val_S(b3,b4) |= b4 & b2,b3 . Val_S(b3,b5) |= b5
iff
b2,b3 . Val_S(b3,CQCSub_&(b4,b5)) |= CQCSub_&(b4,b5);
:: SUBLEMMA:th 26
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3, b4 being Element of CQC-Sub-WFF
st b3 `2 = b4 `2 &
(for b5 being Element of Valuations_in b1 holds
b2,b5 |= CQC_Sub b3
iff
b2,b5 . Val_S(b5,b3) |= b3) &
(for b5 being Element of Valuations_in b1 holds
b2,b5 |= CQC_Sub b4
iff
b2,b5 . Val_S(b5,b4) |= b4)
for b5 being Element of Valuations_in b1 holds
b2,b5 |= CQC_Sub CQCSub_&(b3,b4)
iff
b2,b5 . Val_S(b5,CQCSub_&(b3,b4)) |= CQCSub_&(b3,b4);
:: SUBLEMMA:th 27
theorem
for b1 being Element of [:QC-Sub-WFF,bound_QC-variables:]
for b2 being second_Q_comp of b1
st b1 is quantifiable
holds (Sub_All(b1,b2)) `1 = All(b1 `2,b1 `1 `1) &
(Sub_All(b1,b2)) `2 = b2;
:: SUBLEMMA:attrnot 1 => SUBLEMMA:attr 1
definition
let a1 be Element of [:QC-Sub-WFF,bound_QC-variables:];
attr a1 is CQC-WFF-like means
a1 `1 in CQC-Sub-WFF;
end;
:: SUBLEMMA:dfs 5
definiens
let a1 be Element of [:QC-Sub-WFF,bound_QC-variables:];
To prove
a1 is CQC-WFF-like
it is sufficient to prove
thus a1 `1 in CQC-Sub-WFF;
:: SUBLEMMA:def 5
theorem
for b1 being Element of [:QC-Sub-WFF,bound_QC-variables:] holds
b1 is CQC-WFF-like
iff
b1 `1 in CQC-Sub-WFF;
:: SUBLEMMA:exreg 1
registration
cluster CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:];
end;
:: SUBLEMMA:funcnot 8 => SUBLEMMA:func 8
definition
let a1 be Element of CQC-Sub-WFF;
let a2 be Element of bound_QC-variables;
redefine func [a1, a2] -> CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:];
end;
:: SUBLEMMA:funcnot 9 => SUBLEMMA:func 9
definition
let a1 be CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:];
redefine func a1 `1 -> Element of CQC-Sub-WFF;
end;
:: SUBLEMMA:funcnot 10 => SUBLEMMA:func 10
definition
let a1 be CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:];
let a2 be second_Q_comp of a1;
assume a1 is quantifiable;
func CQCSub_All(A1,A2) -> Element of CQC-Sub-WFF equals
Sub_All(a1,a2);
end;
:: SUBLEMMA:def 6
theorem
for b1 being CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:]
for b2 being second_Q_comp of b1
st b1 is quantifiable
holds CQCSub_All(b1,b2) = Sub_All(b1,b2);
:: SUBLEMMA:th 28
theorem
for b1 being CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:]
for b2 being second_Q_comp of b1
st b1 is quantifiable
holds CQCSub_All(b1,b2) is Sub_universal;
:: SUBLEMMA:funcnot 11 => SUBLEMMA:func 11
definition
let a1 be Element of CQC-Sub-WFF;
assume a1 is Sub_universal;
func CQCSub_the_scope_of A1 -> Element of CQC-Sub-WFF equals
Sub_the_scope_of a1;
end;
:: SUBLEMMA:def 7
theorem
for b1 being Element of CQC-Sub-WFF
st b1 is Sub_universal
holds CQCSub_the_scope_of b1 = Sub_the_scope_of b1;
:: SUBLEMMA:funcnot 12 => SUBLEMMA:func 12
definition
let a1 be Element of CQC-Sub-WFF;
let a2 be Element of CQC-WFF;
assume a1 is Sub_universal & a2 = CQC_Sub CQCSub_the_scope_of a1;
func CQCQuant(A1,A2) -> Element of CQC-WFF equals
Quant(a1,a2);
end;
:: SUBLEMMA:def 8
theorem
for b1 being Element of CQC-Sub-WFF
for b2 being Element of CQC-WFF
st b1 is Sub_universal & b2 = CQC_Sub CQCSub_the_scope_of b1
holds CQCQuant(b1,b2) = Quant(b1,b2);
:: SUBLEMMA:th 29
theorem
for b1 being Element of CQC-Sub-WFF
st b1 is Sub_universal
holds CQC_Sub b1 = CQCQuant(b1,CQC_Sub CQCSub_the_scope_of b1);
:: SUBLEMMA:th 30
theorem
for b1 being CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:]
for b2 being second_Q_comp of b1
st b1 is quantifiable
holds CQCSub_the_scope_of CQCSub_All(b1,b2) = b1 `1;
:: SUBLEMMA:th 31
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable
holds CQCSub_the_scope_of CQCSub_All([b2,b1],b3) = b2 &
CQCQuant(CQCSub_All([b2,b1],b3),CQC_Sub CQCSub_the_scope_of CQCSub_All([b2,b1],b3)) = CQCQuant(CQCSub_All([b2,b1],b3),CQC_Sub b2);
:: SUBLEMMA:th 32
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable
holds CQCQuant(CQCSub_All([b2,b1],b3),CQC_Sub b2) = All(S_Bound @ CQCSub_All([b2,b1],b3),CQC_Sub b2);
:: SUBLEMMA:th 33
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
st b1 in proj1 (b4 `2)
holds b3 . ((@ (b4 `2)) . b1) = (b3 . Val_S(b3,b4)) . b1;
:: SUBLEMMA:th 34
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
st b1 in dom @ (b2 `2)
holds (@ (b2 `2)) . b1 is Element of bound_QC-variables;
:: SUBLEMMA:th 35
theorem
[:QC-WFF,vSUB:] c= proj1 QSub;
:: SUBLEMMA:th 36
theorem
for b1 being CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:]
for b2 being second_Q_comp of b1
for b3 being Element of [:QC-Sub-WFF,bound_QC-variables:]
for b4 being second_Q_comp of b3
st b1 is quantifiable & b3 is quantifiable & Sub_All(b1,b2) = Sub_All(b3,b4)
holds b1 `2 = b3 `2 & b2 = b4;
:: SUBLEMMA:th 37
theorem
for b1 being CQC-WFF-like Element of [:QC-Sub-WFF,bound_QC-variables:]
for b2 being second_Q_comp of b1
for b3 being Element of [:QC-Sub-WFF,bound_QC-variables:]
for b4 being second_Q_comp of b3
st b1 is quantifiable & b3 is quantifiable & CQCSub_All(b1,b2) = Sub_All(b3,b4)
holds b1 `2 = b3 `2 & b2 = b4;
:: SUBLEMMA:th 38
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable
holds Sub_the_bound_of CQCSub_All([b2,b1],b3) = b1;
:: SUBLEMMA:th 39
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable &
b1 in proj2 RestrictSub(b1,All(b1,b2 `1),b3)
holds not S_Bound @ CQCSub_All([b2,b1],b3) in proj2 RestrictSub(b1,All(b1,b2 `1),b3) &
not S_Bound @ CQCSub_All([b2,b1],b3) in Bound_Vars (b2 `1);
:: SUBLEMMA:th 40
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable &
not b1 in proj2 RestrictSub(b1,All(b1,b2 `1),b3)
holds not S_Bound @ CQCSub_All([b2,b1],b3) in proj2 RestrictSub(b1,All(b1,b2 `1),b3);
:: SUBLEMMA:th 41
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable
holds not S_Bound @ CQCSub_All([b2,b1],b3) in proj2 RestrictSub(b1,All(b1,b2 `1),b3);
:: SUBLEMMA:th 42
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable
holds b2 `2 = ExpandSub(b1,b2 `1,RestrictSub(b1,All(b1,b2 `1),b3));
:: SUBLEMMA:th 43
theorem
still_not-bound_in VERUM c= Bound_Vars VERUM;
:: SUBLEMMA:th 44
theorem
for b1 being Element of NAT
for b2 being Element of b1 -ary_QC-pred_symbols
for b3 being CQC-variable_list-like QC-variable_list of b1 holds
still_not-bound_in (b2 ! b3) c= Bound_Vars (b2 ! b3);
:: SUBLEMMA:th 45
theorem
for b1 being Element of CQC-WFF
st still_not-bound_in b1 c= Bound_Vars b1
holds still_not-bound_in 'not' b1 c= Bound_Vars 'not' b1;
:: SUBLEMMA:th 46
theorem
for b1, b2 being Element of CQC-WFF
st still_not-bound_in b1 c= Bound_Vars b1 & still_not-bound_in b2 c= Bound_Vars b2
holds still_not-bound_in (b1 '&' b2) c= Bound_Vars (b1 '&' b2);
:: SUBLEMMA:th 47
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
st still_not-bound_in b1 c= Bound_Vars b1
holds still_not-bound_in All(b2,b1) c= Bound_Vars All(b2,b1);
:: SUBLEMMA:th 48
theorem
for b1 being Element of CQC-WFF holds
still_not-bound_in b1 c= Bound_Vars b1;
:: SUBLEMMA:funcnot 13 => SUBLEMMA:func 13
definition
let a1 be non empty set;
let a2 be Element of a1;
let a3 be Element of bound_QC-variables;
func A3 | A2 -> Function-like Relation of bound_QC-variables,a1 equals
a3 .--> a2;
end;
:: SUBLEMMA:def 9
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Element of bound_QC-variables holds
b3 | b2 = b3 .--> b2;
:: SUBLEMMA:th 49
theorem
for b1 being set
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4 being Element of Valuations_in b3
for b5 being Element of b3
st b2 <> b1
holds (b4 . (b2 | b5)) . b1 = b4 . b1;
:: SUBLEMMA:th 50
theorem
for b1, b2 being Element of bound_QC-variables
for b3 being non empty set
for b4 being Element of Valuations_in b3
for b5 being Element of b3
st b1 = b2
holds (b4 . (b1 | b5)) . b2 = b5;
:: SUBLEMMA:th 51
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4 being interpretation of b3
for b5 being Element of Valuations_in b3 holds
b4,b5 |= All(b2,b1)
iff
for b6 being Element of b3 holds
b4,b5 . (b2 | b6) |= b1;
:: SUBLEMMA:funcnot 14 => SUBLEMMA:func 14
definition
let a1 be Element of CQC-Sub-WFF;
let a2 be Element of bound_QC-variables;
let a3 be second_Q_comp of [a1,a2];
let a4 be non empty set;
let a5 be Element of Valuations_in a4;
func NEx_Val(A5,A1,A2,A3) -> Function-like Relation of bound_QC-variables,a4 equals
(@ RestrictSub(a2,All(a2,a1 `1),a3)) * a5;
end;
:: SUBLEMMA:def 10
theorem
for b1 being Element of CQC-Sub-WFF
for b2 being Element of bound_QC-variables
for b3 being second_Q_comp of [b1,b2]
for b4 being non empty set
for b5 being Element of Valuations_in b4 holds
NEx_Val(b5,b1,b2,b3) = (@ RestrictSub(b2,All(b2,b1 `1),b3)) * b5;
:: SUBLEMMA:funcnot 15 => SUBLEMMA:func 15
definition
let a1 be non empty set;
let a2, a3 be Function-like Relation of bound_QC-variables,a1;
redefine func a2 +* a3 -> Function-like Relation of bound_QC-variables,a1;
idempotence;
:: for a1 being non empty set
:: for a2 being Function-like Relation of bound_QC-variables,a1 holds
:: a2 +* a2 = a2;
end;
:: SUBLEMMA:th 52
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable &
b1 in proj2 RestrictSub(b1,All(b1,b2 `1),b3)
holds S_Bound @ CQCSub_All([b2,b1],b3) = x. upVar(RestrictSub(b1,All(b1,b2 `1),b3),b2 `1);
:: SUBLEMMA:th 53
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable &
not b1 in proj2 RestrictSub(b1,All(b1,b2 `1),b3)
holds S_Bound @ CQCSub_All([b2,b1],b3) = b1;
:: SUBLEMMA:th 54
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1]
st [b4,b1] is quantifiable
for b6 being Element of b2 holds
Val_S(b3 . ((S_Bound @ CQCSub_All([b4,b1],b5)) | b6),b4) = (NEx_Val(b3 . ((S_Bound @ CQCSub_All([b4,b1],b5)) | b6),b4,b1,b5)) +* (b1 | b6) &
proj1 RestrictSub(b1,All(b1,b4 `1),b5) misses {b1};
:: SUBLEMMA:th 55
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of Valuations_in b2
for b5 being Element of CQC-Sub-WFF
for b6 being second_Q_comp of [b5,b1]
st [b5,b1] is quantifiable
holds for b7 being Element of b2 holds
b3,(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7)) . Val_S(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7),b5) |= b5
iff
for b7 being Element of b2 holds
b3,(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7)) . ((NEx_Val(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7),b5,b1,b6)) +* (b1 | b7)) |= b5;
:: SUBLEMMA:th 56
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1]
st [b4,b1] is quantifiable
for b6 being Element of b2 holds
NEx_Val(b3 . ((S_Bound @ CQCSub_All([b4,b1],b5)) | b6),b4,b1,b5) = NEx_Val(b3,b4,b1,b5);
:: SUBLEMMA:th 57
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of Valuations_in b2
for b5 being Element of CQC-Sub-WFF
for b6 being second_Q_comp of [b5,b1]
st [b5,b1] is quantifiable
holds for b7 being Element of b2 holds
b3,(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7)) . ((NEx_Val(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7),b5,b1,b6)) +* (b1 | b7)) |= b5
iff
for b7 being Element of b2 holds
b3,(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7)) . ((NEx_Val(b4,b5,b1,b6)) +* (b1 | b7)) |= b5;
:: SUBLEMMA:th 58
theorem
for b1 being FinSequence of QC-variables
st rng b1 c= bound_QC-variables
holds still_not-bound_in b1 = rng b1;
:: SUBLEMMA:th 59
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of b2 holds
dom b3 = bound_QC-variables & dom (b1 | b4) = {b1};
:: SUBLEMMA:th 60
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being CQC-variable_list-like QC-variable_list of b1 holds
b3 *' b4 = b4 * (b3 | still_not-bound_in b4);
:: SUBLEMMA:th 61
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of b1 -ary_QC-pred_symbols
for b5 being CQC-variable_list-like QC-variable_list of b1
for b6, b7 being Element of Valuations_in b2
st b6 | still_not-bound_in (b4 ! b5) = b7 | still_not-bound_in (b4 ! b5)
holds b3,b6 |= b4 ! b5
iff
b3,b7 |= b4 ! b5;
:: SUBLEMMA:th 62
theorem
for b1 being Element of CQC-WFF
for b2 being non empty set
for b3 being interpretation of b2
st for b4, b5 being Element of Valuations_in b2
st b4 | still_not-bound_in b1 = b5 | still_not-bound_in b1
holds b3,b4 |= b1
iff
b3,b5 |= b1
for b4, b5 being Element of Valuations_in b2
st b4 | still_not-bound_in 'not' b1 = b5 | still_not-bound_in 'not' b1
holds b3,b4 |= 'not' b1
iff
b3,b5 |= 'not' b1;
:: SUBLEMMA:th 63
theorem
for b1, b2 being Element of CQC-WFF
for b3 being non empty set
for b4 being interpretation of b3
st (for b5, b6 being Element of Valuations_in b3
st b5 | still_not-bound_in b1 = b6 | still_not-bound_in b1
holds b4,b5 |= b1
iff
b4,b6 |= b1) &
(for b5, b6 being Element of Valuations_in b3
st b5 | still_not-bound_in b2 = b6 | still_not-bound_in b2
holds b4,b5 |= b2
iff
b4,b6 |= b2)
for b5, b6 being Element of Valuations_in b3
st b5 | still_not-bound_in (b1 '&' b2) = b6 | still_not-bound_in (b1 '&' b2)
holds b4,b5 |= b1 '&' b2
iff
b4,b6 |= b1 '&' b2;
:: SUBLEMMA:th 64
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of b2
for b5 being set
st b5 c= bound_QC-variables
holds dom (b3 | b5) = dom ((b3 . (b1 | b4)) | b5) &
dom (b3 | b5) = b5;
:: SUBLEMMA:th 65
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4, b5 being Element of Valuations_in b3
for b6 being Element of b3
st b4 | still_not-bound_in b1 = b5 | still_not-bound_in b1
holds (b4 . (b2 | b6)) | still_not-bound_in b1 = (b5 . (b2 | b6)) | still_not-bound_in b1;
:: SUBLEMMA:th 66
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
still_not-bound_in b1 c= (still_not-bound_in All(b2,b1)) \/ {b2};
:: SUBLEMMA:th 67
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4, b5 being Element of Valuations_in b3
for b6 being Element of b3
st b4 | ((still_not-bound_in b1) \ {b2}) = b5 | ((still_not-bound_in b1) \ {b2})
holds (b4 . (b2 | b6)) | still_not-bound_in b1 = (b5 . (b2 | b6)) | still_not-bound_in b1;
:: SUBLEMMA:th 68
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4 being interpretation of b3
st for b5, b6 being Element of Valuations_in b3
st b5 | still_not-bound_in b1 = b6 | still_not-bound_in b1
holds b4,b5 |= b1
iff
b4,b6 |= b1
for b5, b6 being Element of Valuations_in b3
st b5 | still_not-bound_in All(b2,b1) = b6 | still_not-bound_in All(b2,b1)
holds b4,b5 |= All(b2,b1)
iff
b4,b6 |= All(b2,b1);
:: SUBLEMMA:th 70
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of CQC-WFF
for b4, b5 being Element of Valuations_in b1
st b4 | still_not-bound_in b3 = b5 | still_not-bound_in b3
holds b2,b4 |= b3
iff
b2,b5 |= b3;
:: SUBLEMMA:th 71
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1]
for b6 being Element of b2
st [b4,b1] is quantifiable
holds ((b3 . ((S_Bound @ CQCSub_All([b4,b1],b5)) | b6)) . ((NEx_Val(b3,b4,b1,b5)) +* (b1 | b6))) | still_not-bound_in (b4 `1) = (b3 . ((NEx_Val(b3,b4,b1,b5)) +* (b1 | b6))) | still_not-bound_in (b4 `1);
:: SUBLEMMA:th 72
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of Valuations_in b2
for b5 being Element of CQC-Sub-WFF
for b6 being second_Q_comp of [b5,b1]
st [b5,b1] is quantifiable
holds for b7 being Element of b2 holds
b3,(b4 . ((S_Bound @ CQCSub_All([b5,b1],b6)) | b7)) . ((NEx_Val(b4,b5,b1,b6)) +* (b1 | b7)) |= b5
iff
for b7 being Element of b2 holds
b3,b4 . ((NEx_Val(b4,b5,b1,b6)) +* (b1 | b7)) |= b5;
:: SUBLEMMA:th 73
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1] holds
dom NEx_Val(b3,b4,b1,b5) = proj1 RestrictSub(b1,All(b1,b4 `1),b5);
:: SUBLEMMA:th 75
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of Valuations_in b2
for b5 being Element of CQC-Sub-WFF
for b6 being second_Q_comp of [b5,b1]
st [b5,b1] is quantifiable
holds for b7 being Element of b2 holds
b3,b4 . ((NEx_Val(b4,b5,b1,b6)) +* (b1 | b7)) |= b5
iff
for b7 being Element of b2 holds
b3,(b4 . NEx_Val(b4,b5,b1,b6)) . (b1 | b7) |= b5;
:: SUBLEMMA:th 76
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of Valuations_in b2
for b5 being Element of CQC-Sub-WFF
for b6 being second_Q_comp of [b5,b1] holds
for b7 being Element of b2 holds
b3,(b4 . NEx_Val(b4,b5,b1,b6)) . (b1 | b7) |= b5
iff
for b7 being Element of b2 holds
b3,(b4 . NEx_Val(b4,b5,b1,b6)) . (b1 | b7) |= b5 `1;
:: SUBLEMMA:th 78
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being CQC-variable_list-like QC-variable_list of b1
for b4 being Element of Valuations_in b2
for b5, b6, b7 being Function-like Relation of bound_QC-variables,b2
st (for b8 being Element of bound_QC-variables
st b8 in dom b6
holds not b8 in still_not-bound_in b3) &
(for b8 being Element of bound_QC-variables
st b8 in dom b7
holds b7 . b8 = b4 . b8) &
dom b5 misses dom b7
holds (b4 . b5) *' b3 = (b4 . ((b5 +* b6) +* b7)) *' b3;
:: SUBLEMMA:th 79
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of b1 -ary_QC-pred_symbols
for b5 being CQC-variable_list-like QC-variable_list of b1
for b6 being Element of Valuations_in b2
for b7, b8, b9 being Function-like Relation of bound_QC-variables,b2
st (for b10 being Element of bound_QC-variables
st b10 in dom b8
holds not b10 in still_not-bound_in (b4 ! b5)) &
(for b10 being Element of bound_QC-variables
st b10 in dom b9
holds b9 . b10 = b6 . b10) &
dom b7 misses dom b9
holds b3,b6 . b7 |= b4 ! b5
iff
b3,b6 . ((b7 +* b8) +* b9) |= b4 ! b5;
:: SUBLEMMA:th 80
theorem
for b1 being Element of CQC-WFF
for b2 being non empty set
for b3 being interpretation of b2
st for b4 being Element of Valuations_in b2
for b5, b6, b7 being Function-like Relation of bound_QC-variables,b2
st (for b8 being Element of bound_QC-variables
st b8 in dom b6
holds not b8 in still_not-bound_in b1) &
(for b8 being Element of bound_QC-variables
st b8 in dom b7
holds b7 . b8 = b4 . b8) &
dom b5 misses dom b7
holds b3,b4 . b5 |= b1
iff
b3,b4 . ((b5 +* b6) +* b7) |= b1
for b4 being Element of Valuations_in b2
for b5, b6, b7 being Function-like Relation of bound_QC-variables,b2
st (for b8 being Element of bound_QC-variables
st b8 in dom b6
holds not b8 in still_not-bound_in 'not' b1) &
(for b8 being Element of bound_QC-variables
st b8 in dom b7
holds b7 . b8 = b4 . b8) &
dom b5 misses dom b7
holds b3,b4 . b5 |= 'not' b1
iff
b3,b4 . ((b5 +* b6) +* b7) |= 'not' b1;
:: SUBLEMMA:th 81
theorem
for b1, b2 being Element of CQC-WFF
for b3 being non empty set
for b4 being interpretation of b3
st (for b5 being Element of Valuations_in b3
for b6, b7, b8 being Function-like Relation of bound_QC-variables,b3
st (for b9 being Element of bound_QC-variables
st b9 in dom b7
holds not b9 in still_not-bound_in b1) &
(for b9 being Element of bound_QC-variables
st b9 in dom b8
holds b8 . b9 = b5 . b9) &
dom b6 misses dom b8
holds b4,b5 . b6 |= b1
iff
b4,b5 . ((b6 +* b7) +* b8) |= b1) &
(for b5 being Element of Valuations_in b3
for b6, b7, b8 being Function-like Relation of bound_QC-variables,b3
st (for b9 being Element of bound_QC-variables
st b9 in dom b7
holds not b9 in still_not-bound_in b2) &
(for b9 being Element of bound_QC-variables
st b9 in dom b8
holds b8 . b9 = b5 . b9) &
dom b6 misses dom b8
holds b4,b5 . b6 |= b2
iff
b4,b5 . ((b6 +* b7) +* b8) |= b2)
for b5 being Element of Valuations_in b3
for b6, b7, b8 being Function-like Relation of bound_QC-variables,b3
st (for b9 being Element of bound_QC-variables
st b9 in dom b7
holds not b9 in still_not-bound_in (b1 '&' b2)) &
(for b9 being Element of bound_QC-variables
st b9 in dom b8
holds b8 . b9 = b5 . b9) &
dom b6 misses dom b8
holds b4,b5 . b6 |= b1 '&' b2
iff
b4,b5 . ((b6 +* b7) +* b8) |= b1 '&' b2;
:: SUBLEMMA:th 82
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4 being Function-like Relation of bound_QC-variables,b3
st for b5 being Element of bound_QC-variables
st b5 in dom b4
holds not b5 in still_not-bound_in All(b2,b1)
for b5 being Element of bound_QC-variables
st b5 in (dom b4) \ {b2}
holds not b5 in still_not-bound_in b1;
:: SUBLEMMA:th 83
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Function-like Relation of bound_QC-variables,b2
for b5 being Relation-like Function-like set
st (for b6 being Element of bound_QC-variables
st b6 in proj1 b5
holds b5 . b6 = b3 . b6) &
dom b4 misses proj1 b5
for b6 being Element of bound_QC-variables
st b6 in (proj1 b5) \ {b1}
holds (b5 | ((proj1 b5) \ {b1})) . b6 = (b3 . b4) . b6;
:: SUBLEMMA:th 84
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being non empty set
for b4 being interpretation of b3
st for b5 being Element of Valuations_in b3
for b6, b7, b8 being Function-like Relation of bound_QC-variables,b3
st (for b9 being Element of bound_QC-variables
st b9 in dom b7
holds not b9 in still_not-bound_in b1) &
(for b9 being Element of bound_QC-variables
st b9 in dom b8
holds b8 . b9 = b5 . b9) &
dom b6 misses dom b8
holds b4,b5 . b6 |= b1
iff
b4,b5 . ((b6 +* b7) +* b8) |= b1
for b5 being Element of Valuations_in b3
for b6, b7, b8 being Function-like Relation of bound_QC-variables,b3
st (for b9 being Element of bound_QC-variables
st b9 in dom b7
holds not b9 in still_not-bound_in All(b2,b1)) &
(for b9 being Element of bound_QC-variables
st b9 in dom b8
holds b8 . b9 = b5 . b9) &
dom b6 misses dom b8
holds b4,b5 . b6 |= All(b2,b1)
iff
b4,b5 . ((b6 +* b7) +* b8) |= All(b2,b1);
:: SUBLEMMA:th 85
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of CQC-WFF
for b4 being Element of Valuations_in b1
for b5, b6, b7 being Function-like Relation of bound_QC-variables,b1
st (for b8 being Element of bound_QC-variables
st b8 in dom b6
holds not b8 in still_not-bound_in b3) &
(for b8 being Element of bound_QC-variables
st b8 in dom b7
holds b7 . b8 = b4 . b8) &
dom b5 misses dom b7
holds b2,b4 . b5 |= b3
iff
b2,b4 . ((b5 +* b6) +* b7) |= b3;
:: SUBLEMMA:funcnot 16 => SUBLEMMA:func 16
definition
let a1 be Element of CQC-WFF;
func RSub1 A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Element of bound_QC-variables st
b2 = b1 & not b2 in still_not-bound_in a1;
end;
:: SUBLEMMA:def 11
theorem
for b1 being Element of CQC-WFF
for b2 being set holds
b2 = RSub1 b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being Element of bound_QC-variables st
b4 = b3 & not b4 in still_not-bound_in b1;
:: SUBLEMMA:funcnot 17 => SUBLEMMA:func 17
definition
let a1 be Element of CQC-WFF;
let a2 be Element of vSUB;
func RSub2(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Element of bound_QC-variables st
b2 = b1 & b2 in still_not-bound_in a1 & b2 = (@ a2) . b2;
end;
:: SUBLEMMA:def 12
theorem
for b1 being Element of CQC-WFF
for b2 being Element of vSUB
for b3 being set holds
b3 = RSub2(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Element of bound_QC-variables st
b5 = b4 & b5 in still_not-bound_in b1 & b5 = (@ b2) . b5;
:: SUBLEMMA:th 86
theorem
for b1 being Element of CQC-WFF
for b2 being Element of vSUB holds
dom ((@ b2) | RSub1 b1) misses dom ((@ b2) | RSub2(b1,b2));
:: SUBLEMMA:th 87
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being Element of vSUB holds
@ RestrictSub(b2,All(b2,b1),b3) = (@ b3) \ (((@ b3) | RSub1 All(b2,b1)) +* ((@ b3) | RSub2(All(b2,b1),b3)));
:: SUBLEMMA:th 88
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
for b3 being Element of vSUB holds
dom @ RestrictSub(b2,b1,b3) misses (dom ((@ b3) | RSub1 b1)) \/ dom ((@ b3) | RSub2(b1,b3));
:: SUBLEMMA:th 89
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of CQC-Sub-WFF
for b3 being second_Q_comp of [b2,b1]
st [b2,b1] is quantifiable
holds @ ((CQCSub_All([b2,b1],b3)) `2) = ((@ RestrictSub(b1,All(b1,b2 `1),b3)) +* ((@ b3) | RSub1 All(b1,b2 `1))) +* ((@ b3) | RSub2(All(b1,b2 `1),b3));
:: SUBLEMMA:th 90
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being Element of Valuations_in b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1]
st [b4,b1] is quantifiable
holds ex b6, b7 being Function-like Relation of bound_QC-variables,b2 st
(for b8 being Element of bound_QC-variables
st b8 in dom b6
holds not b8 in still_not-bound_in All(b1,b4 `1)) &
(for b8 being Element of bound_QC-variables
st b8 in dom b7
holds b7 . b8 = b3 . b8) &
dom NEx_Val(b3,b4,b1,b5) misses dom b7 &
b3 . Val_S(b3,CQCSub_All([b4,b1],b5)) = b3 . (((NEx_Val(b3,b4,b1,b5)) +* b6) +* b7);
:: SUBLEMMA:th 91
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1]
st [b4,b1] is quantifiable
for b6 being Element of Valuations_in b2 holds
b3,b6 . NEx_Val(b6,b4,b1,b5) |= All(b1,b4 `1)
iff
b3,b6 . Val_S(b6,CQCSub_All([b4,b1],b5)) |= CQCSub_All([b4,b1],b5);
:: SUBLEMMA:th 92
theorem
for b1 being Element of bound_QC-variables
for b2 being non empty set
for b3 being interpretation of b2
for b4 being Element of CQC-Sub-WFF
for b5 being second_Q_comp of [b4,b1]
st [b4,b1] is quantifiable &
(for b6 being Element of Valuations_in b2 holds
b3,b6 |= CQC_Sub b4
iff
b3,b6 . Val_S(b6,b4) |= b4)
for b6 being Element of Valuations_in b2 holds
b3,b6 |= CQC_Sub CQCSub_All([b4,b1],b5)
iff
b3,b6 . Val_S(b6,CQCSub_All([b4,b1],b5)) |= CQCSub_All([b4,b1],b5);
:: SUBLEMMA:sch 1
scheme SUBLEMMA:sch 1
for b1 being Element of CQC-Sub-WFF holds
P1[b1]
provided
for b1, b2 being Element of CQC-Sub-WFF
for b3 being Element of bound_QC-variables
for b4 being second_Q_comp of [b1,b3]
for b5 being Element of NAT
for b6 being CQC-variable_list-like QC-variable_list of b5
for b7 being Element of b5 -ary_QC-pred_symbols
for b8 being Element of vSUB holds
P1[Sub_P(b7,b6,b8)] &
(b1 is Sub_VERUM implies P1[b1]) &
(P1[b1] implies P1[Sub_not b1]) &
(b1 `2 = b2 `2 & P1[b1] & P1[b2] implies P1[CQCSub_&(b1,b2)]) &
([b1,b3] is quantifiable & P1[b1] implies P1[CQCSub_All([b1,b3],b4)]);
:: SUBLEMMA:th 93
theorem
for b1 being non empty set
for b2 being interpretation of b1
for b3 being Element of CQC-Sub-WFF
for b4 being Element of Valuations_in b1 holds
b2,b4 |= CQC_Sub b3
iff
b2,b4 . Val_S(b4,b3) |= b3;