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 Th3:
  p ||= r implies PP_and(p,q) ||= r
  proof
    set F = PP_and(p,q);
A1: dom(F) = {d where d is Element of D:
     d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
     or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE} by PARTPR_1:16;
    assume
A2: p ||= r;
    let d such that
A3: d in dom F and
A4: F.d = TRUE;
    consider d1 being Element of D such that
A5: d = d1 and
A6: d1 in dom p & p.d1 = FALSE or d1 in dom q & q.d1 = FALSE
    or d1 in dom p & p.d1 = TRUE & d1 in dom q & q.d1 = TRUE by A1,A3;
    per cases by A6;
    suppose d1 in dom p & p.d1 = FALSE;
      hence thesis by A4,A5,PARTPR_1:19;
    end;
    suppose d1 in dom q & q.d1 = FALSE;
      hence thesis by A4,A5,PARTPR_1:19;
    end;
    suppose d1 in dom p & d1 in dom q & p.d1 = TRUE & q.d1 = TRUE;
      hence thesis by A2,A5;
    end;
  end;
