reserve v,x for object;
reserve D,V,A for set;
reserve n for Nat;
reserve p,q for PartialPredicate of D;
reserve f,g for BinominativeFunction of D;
reserve D for non empty set;
reserve d for Element of D;
reserve f,g for BinominativeFunction of D;
reserve p,q,r,s for PartialPredicate of D;

theorem Th6:
  PP_or(p,q) ||= r implies p ||= r
  proof
    set F = PP_or(p,q);
A1: dom(F) = {d where d is Element of D:
     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;
    assume
A2: F ||= r;
    let d;
    assume d in dom p & p.d = TRUE;
    then d in dom F & F.d = TRUE by A1,PARTPR_1:def 4;
    hence thesis by A2;
  end;
