let V, A be set ; for p, q being SCPartialNominativePredicate of V,A holds dom (PP_or (p,q)) = { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
let p, q be SCPartialNominativePredicate of V,A; dom (PP_or (p,q)) = { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
set X = { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } ;
set Y = { d where d is Element of ND (V,A) : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } ;
{ d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } = { d where d is Element of ND (V,A) : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
proof
thus
{ d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } c= { d where d is Element of ND (V,A) : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
XBOOLE_0:def 10 { d where d is Element of ND (V,A) : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } c= { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
let x be
object ;
TARSKI:def 3 ( not x in { d where d is Element of ND (V,A) : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } or x in { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } )
assume
x in { d where d is Element of ND (V,A) : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
;
x in { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
then consider d being
Element of
ND (
V,
A)
such that A1:
(
d = x & ( (
d in dom p &
p . d = TRUE ) or (
d in dom q &
q . d = TRUE ) or (
d in dom p &
p . d = FALSE &
d in dom q &
q . d = FALSE ) ) )
;
d is
TypeSCNominativeData of
V,
A
by NOMIN_1:39;
hence
x in { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
by A1;
verum
end;
hence
dom (PP_or (p,q)) = { d where d is TypeSCNominativeData of V,A : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) }
by PARTPR_1:def 4; verum