theorem
  id(X) +* id(Y) = id(X \/ Y)
proof
A1: for x being object holds
x in dom id(X \/ Y) implies (id(X) +* id(Y)).x = id(X \/ Y).x
  proof let x be object;
    assume
A2: x in dom id(X \/ Y);
    now
      per cases;
      suppose
A3:     x in Y;
        dom id Y = Y;
        hence (id(X) +* id(Y)).x = (id Y).x by A3,Th13
          .= x by A3,FUNCT_1:18
          .= id(X \/ Y).x by A2,FUNCT_1:18;
      end;
      suppose
A4:     not x in Y;
        then
A5:     x in X by A2,XBOOLE_0:def 3;
        not x in dom id Y by A4;
        hence (id(X) +* id(Y)).x = (id X).x by Th11
          .= x by A5,FUNCT_1:18
          .= id(X \/ Y).x by A2,FUNCT_1:18;
      end;
    end;
    hence thesis;
  end;
  dom(id(X) +* id(Y)) = dom id X \/ dom id Y by Def1
    .= X \/ dom id Y
    .= X \/ Y
    .= dom id(X \/ Y);
  hence thesis by A1;
end;
