
theorem Th4:
  for I being non empty set, a,b,c,d being set, i,j being Element of I
  holds c in ((i-singleton a)(\/)(j-singleton d)).b iff
  b = i & c = a or b = j & c = d
  proof
    let I be non empty set;
    let a,b,c,d be set;
    let i,j be Element of I;
    hereby
      assume A1: c in ((i-singleton a)(\/)(j-singleton d)).b;
      assume A2: not (b = i & c = a);
      b in dom((i-singleton a)(\/)(j-singleton d)) by A1,FUNCT_1:def 2;
      then b in I by PARTFUN1:def 2;
      then c in ((i-singleton a).b)\/((j-singleton d).b) by A1,PBOOLE:def 4;
      then c in (i-singleton a).b or c in (j-singleton d).b by XBOOLE_0:def 3;
      hence b = j & c = d by A2,Th3;
    end;
    assume A3: b = i & c = a or b = j & c = d;
    then c in (i-singleton a).b or c in (j-singleton d).b by Th3;
    then c in ((i-singleton a).b)\/((j-singleton d).b) by XBOOLE_0:def 3;
    hence c in ((i-singleton a)(\/)(j-singleton d)).b by A3,PBOOLE:def 4;
  end;
