reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;

theorem Th57:
  PP_or(PP_BottomPred(D),PP_True(D)) = PP_True(D)
  proof
    set B = PP_BottomPred(D);
    set T = PP_True(D);
    set o = PP_or(B,T);
A1: dom(o) = {d where d is Element of D:
    d in dom B & B.d = TRUE or d in dom T & T.d = TRUE
    or d in dom B & B.d = FALSE & d in dom T & T.d = FALSE} by Def4;
    thus dom o = dom T
    proof
      thus dom o c= dom T;
      thus dom T c= dom o
      proof
        let x;
        assume
A2:     x in dom T;
        then T.x = TRUE by FUNCOP_1:7;
        hence thesis by A1,A2;
      end;
    end;
    let x;
    assume x in dom o;
    then per cases by Th8;
    suppose x in dom B & B.x = TRUE or
      x in dom B & B.x = FALSE & x in dom T & T.x = FALSE;
      hence thesis;
    end;
    suppose x in dom T & T.x = TRUE;
      hence thesis by Def4;
    end;
  end;
