let V, A be set ; :: thesis: 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; :: thesis: 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 ) ) } :: according to XBOOLE_0:def 10 :: thesis: { 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 ) ) }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not 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 ) ) } or 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 ) ) } )
assume 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 ) ) } ; :: thesis: 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 ) ) }
then ex d being TypeSCNominativeData of V,A st
( 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 ) ) ) ;
hence 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 ) ) } ; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( 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 ) ) } ; :: thesis: 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; :: thesis: 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; :: thesis: verum