 reserve X, Y for set, A for Ordinal;

theorem
  succRel(X /\ Y) = succRel(X) /\ succRel(Y)
proof
  now
    let z be object;
    hereby
      assume A1: z in succRel(X /\ Y);
      then consider x,y being object such that
        A2: z = [x,y] by RELAT_1:def 1;
      reconsider a=x,b=y as set by TARSKI:1;
      [a,b] in succRel(X /\ Y) by A1, A2;
      then a in X /\ Y & b in X /\ Y & b = succ a by Def1;
      then a in X & b in X & a in Y & b in Y & b = succ a by XBOOLE_0:def 4;
      hence z in succRel(X) & z in succRel(Y) by A2, Def1;
    end;
    assume A3: z in succRel(X) & z in succRel(Y);
    then consider x,y being object such that
      A4: z = [x,y] by RELAT_1:def 1;
    reconsider a=x,b=y as set by TARSKI:1;
    [a,b] in succRel(X) & [a,b] in succRel(Y) by A3, A4;
    then a in X & b in X & a in Y & b in Y & b = succ a by Def1;
    then a in X /\ Y & b in X /\ Y & b = succ a by XBOOLE_0:def 4;
    hence z in succRel(X /\ Y) by A4, Def1;
  end;
  hence thesis by XBOOLE_0:def 4;
end;
